🧠:<think>Alright, so I've got this problem about an author using graph theory to structure her novel. It's divided into two parts, and I need to figure out both. Let me take it step by step.Problem 1: The author has 10 main characters, each represented as a vertex in a graph. She wants a connected graph where each character has a unique relationship with every other character, and it's Hamiltonian, meaning there's a Hamiltonian cycle. I need to find how many distinct Hamiltonian cycles can be formed in this graph.Hmm, okay. So, first, let's clarify what a Hamiltonian cycle is. It's a cycle that visits every vertex exactly once and returns to the starting vertex. In a complete graph with n vertices, the number of Hamiltonian cycles is (n-1)! / 2. Wait, why? Because in a complete graph, every permutation of the vertices (except for rotations and reflections) gives a unique cycle.But wait, the problem says the graph is connected and each character has a unique relationship with every other character. Does that mean it's a complete graph? Because in a complete graph, every pair of vertices is connected by an edge. So, if each character has a unique relationship with every other, that sounds like a complete graph.So, for a complete graph with 10 vertices, the number of Hamiltonian cycles would be (10-1)! / 2 = 9! / 2. Let me compute that.9! is 362880. Divided by 2 is 181440. So, 181,440 distinct Hamiltonian cycles.Wait, but hold on. The problem says "the graph is Hamiltonian." Does that mean it's just any Hamiltonian graph, not necessarily complete? Because a Hamiltonian graph is one that has at least one Hamiltonian cycle, but it doesn't have to be complete.But the problem also says "each character has a unique relationship with every other character." That seems to imply that every pair of characters is connected, i.e., a complete graph. So, I think it's safe to assume it's a complete graph.Therefore, the number of distinct Hamiltonian cycles is 9! / 2 = 181,440.Problem 2: Now, the author introduces a subplot with a secret society of 4 characters. She wants their interactions to form a complete subgraph. Given the original 10-character graph, how many different ways can this 4-character secret society be chosen and arranged as a complete subgraph within the Hamiltonian graph?So, first, I need to choose 4 characters out of 10. The number of ways to choose 4 is C(10,4). Then, within this subgroup, the interactions form a complete subgraph, which is a K4. But wait, in the original graph, which is a complete graph, every subset of 4 vertices already forms a complete subgraph. So, the number of complete subgraphs of size 4 is just the number of ways to choose 4 vertices, which is C(10,4).But wait, the question says "chosen and arranged as a complete subgraph within the Hamiltonian graph." Hmm, does that mean something else? Maybe considering the arrangement in the context of the Hamiltonian cycle?Wait, the original graph is a complete graph, which is Hamiltonian. So, any subset of 4 vertices will form a complete subgraph, regardless of the Hamiltonian cycle. So, perhaps the number is just C(10,4).But let me think again. Maybe the question is asking for the number of ways to choose the 4 characters and then arrange them in the Hamiltonian cycle such that their interactions form a complete subgraph.Wait, but in a complete graph, any 4 vertices already form a complete subgraph, so regardless of how they are arranged in the Hamiltonian cycle, their interactions are all present. So, perhaps the number is just the number of ways to choose 4 characters, which is C(10,4).But let's compute that. C(10,4) is 210. So, 210 different ways.Wait, but maybe the question is more about embedding the K4 into the Hamiltonian cycle. But in a complete graph, every set of 4 vertices is a complete subgraph, so regardless of the Hamiltonian cycle, the interactions are already complete. So, the number of ways is just the number of 4-vertex subsets, which is 210.Alternatively, if the graph wasn't complete, but just Hamiltonian, then the number of complete subgraphs of size 4 would depend on the structure. But since the original graph is complete, it's just C(10,4).So, I think the answer is 210.But wait, let me make sure. The problem says "within the Hamiltonian graph." So, perhaps it's considering the Hamiltonian cycle as the structure, and within that, how many K4s are there.But in a Hamiltonian cycle, which is a cycle graph, the number of complete subgraphs of size 4 is zero because a cycle doesn't contain any complete subgraphs larger than triangles, and even triangles are not present unless it's a complete graph.Wait, now I'm confused. The original graph is a complete graph, which is Hamiltonian. So, the Hamiltonian cycle is just one cycle in the complete graph. But the complete graph has many more edges beyond the Hamiltonian cycle.So, the secret society's interactions form a complete subgraph, which is K4. So, in the complete graph, the number of K4s is C(10,4). But if we're considering within the Hamiltonian graph, which is just the cycle, then the number of K4s is zero because a cycle doesn't have enough edges.Wait, but the problem says "within the Hamiltonian graph." Hmm. Maybe it's a bit ambiguous.Wait, the original graph is a complete graph, which is Hamiltonian. So, the Hamiltonian graph here refers to the complete graph, which is Hamiltonian. So, the complete graph itself is the Hamiltonian graph.Therefore, the number of K4s is C(10,4) = 210.Alternatively, if the Hamiltonian graph was just the cycle, then it's different, but since the complete graph is Hamiltonian, and the question is about the original 10-character graph, which is complete, then the number is 210.So, I think the answer is 210.But let me think again. The problem says "within the Hamiltonian graph." So, perhaps it's considering the Hamiltonian cycle as the graph, not the complete graph. So, if the graph is just the Hamiltonian cycle (a cycle with 10 vertices), then the number of complete subgraphs of size 4 is zero because a cycle doesn't have any complete subgraphs beyond edges.But that contradicts the earlier part where the graph is complete. So, maybe the graph is complete, which is Hamiltonian, and the question is about the complete graph, not just the cycle.So, I think the answer is 210.Wait, another angle: the author wants the interactions within the subgroup to form a complete subgraph. So, regardless of the Hamiltonian cycle, the number of ways is just the number of 4-vertex subsets, which is C(10,4) = 210.Yes, that makes sense.So, to summarize:1. The number of Hamiltonian cycles in a complete graph with 10 vertices is 9! / 2 = 181,440.2. The number of ways to choose a 4-character secret society that forms a complete subgraph is C(10,4) = 210.I think that's it.</think>

🧠:<think>Okay, so I have this problem where Alex had to move because their property was sold, and now they want to analyze their rental expenses over 10 years. They rented three different apartments, each with their own rental cost functions. The goal is to figure out the total rent paid over the 10 years and then calculate the expected moving expenses for the next 10 years considering a probability of moving each year.Let me start with the first sub-problem: calculating the total rent paid over 10 years. Alex rented the first apartment for 5 years, the second for 3 years, and the third for 2 years. So, the total time adds up to 10 years, which makes sense.First, I need to understand each rental function:1. The first apartment's rent is given by ( f(t) = 800 + 20t ), where t is the number of months since Alex started renting the first apartment. So, this rent increases by 20 every month.2. The second apartment's rent is ( g(t) = 1200 + 10(t - 60) ) for ( t geq 60 ). Since Alex moved to the second apartment after 5 years, which is 60 months, the rent starts increasing from that point. So, it's 1200 plus 10 per month after the 60th month.3. The third apartment's rent is ( h(t) = 1500 + 5(t - 120) ) for ( t geq 120 ). Alex moved here after 8 years (which is 96 months, but wait, 5 + 3 = 8 years, so 96 months? Hmm, but the function starts at t=120. Wait, that seems contradictory. Let me think.Wait, hold on. The first apartment was for 5 years, which is 60 months. Then the second for 3 years, which is 36 months, so total 96 months. Then the third apartment for 2 years, which is 24 months, totaling 120 months, which is 10 years. So, the third apartment starts at t=96, but the function is defined for t >= 120. That seems inconsistent. Maybe I need to adjust the functions or the periods.Wait, perhaps the functions are defined relative to when each apartment was rented, not from the start. Let me check the problem statement again.It says, "where ( f(t) ), ( g(t) ), and ( h(t) ) represent the monthly rental costs... and ( t ) represents the number of months since Alex started renting the first apartment." So, t is measured from the start of the first apartment. Therefore, the second apartment is rented starting at t=60, and the third at t=96.But the functions for the second and third apartments are defined as:( g(t) = 1200 + 10(t - 60) ) for ( t geq 60 )( h(t) = 1500 + 5(t - 120) ) for ( t geq 120 )Wait, so for the third apartment, the function starts at t=120, but Alex started renting it at t=96. That seems like a problem. Maybe the functions are defined relative to when each apartment was started? Or perhaps there's a typo in the problem.Wait, let me read the problem again:"Alex rented three different apartments over a period of 10 years. The rental prices of the apartments can be modeled by the following functions, where ( f(t) ), ( g(t) ), and ( h(t) ) represent the monthly rental costs (in dollars) of the first, second, and third apartments, respectively, and ( t ) represents the number of months since Alex started renting the first apartment:1. ( f(t) = 800 + 20t )2. ( g(t) = 1200 + 10(t - 60) ) for ( t geq 60 )3. ( h(t) = 1500 + 5(t - 120) ) for ( t geq 120 )"So, t is the number of months since the first apartment was rented. So, for the second apartment, which was rented at t=60, the function is defined as 1200 + 10(t - 60) for t >=60. That makes sense because at t=60, the rent is 1200, and then increases by 10 each month after that.Similarly, the third apartment was rented at t=96 (since 5 years is 60 months, plus 3 years is 36 months, so 60+36=96). But the function h(t) is defined for t >=120, which is 10 years later. That seems odd because Alex only rented the third apartment for 2 years, which is 24 months, starting at t=96, so it should go up to t=120. So, the function h(t) is defined for t >=120, but Alex only rented it from t=96 to t=120. So, perhaps the function is defined for t >=120, but Alex only used it from t=96 to t=120. That seems conflicting.Wait, maybe I need to adjust the functions. Perhaps the functions are meant to be applied only during the time Alex was renting each apartment. So, for the first apartment, t from 0 to 60, second from 60 to 96, third from 96 to 120.But the functions are defined as:f(t) = 800 +20t for all t, but only relevant from t=0 to t=60.g(t) = 1200 +10(t -60) for t >=60, so from t=60 onwards, but Alex only rented it until t=96.h(t) =1500 +5(t -120) for t >=120, but Alex only rented it until t=120. Wait, that would mean that h(t) is only applicable at t=120, but Alex started renting it at t=96.This seems inconsistent. Maybe the functions are defined with t as the number of months since each apartment was rented. That is, for the second apartment, t=0 corresponds to when Alex started renting it, which was at t=60 in the overall timeline. Similarly, for the third apartment, t=0 corresponds to t=96 in the overall timeline.If that's the case, then we need to adjust the functions accordingly.Wait, the problem says: "t represents the number of months since Alex started renting the first apartment." So, t is consistent across all functions. Therefore, for the second apartment, which was rented at t=60, the function g(t) is 1200 +10(t -60) for t >=60. So, that works because when t=60, g(t)=1200, and then increases by 10 each month.Similarly, for the third apartment, which was rented at t=96, the function h(t) is 1500 +5(t -120) for t >=120. Wait, but Alex started renting it at t=96, so the function should be defined for t >=96, not 120. That seems like a mistake in the problem statement.Alternatively, maybe the functions are defined with t as the number of months since each apartment was rented. That is, for the second apartment, t=0 is when Alex started renting it, which was at t=60 in the overall timeline. So, the function g(t) would be 1200 +10t, but in terms of the overall timeline, it's 1200 +10(t -60). Similarly, for the third apartment, t=0 is when Alex started renting it, which was at t=96, so h(t) would be 1500 +5t, but in terms of the overall timeline, it's 1500 +5(t -96). However, the problem states h(t) =1500 +5(t -120) for t >=120, which is inconsistent.This is confusing. Maybe I need to proceed with the given functions as they are, assuming that perhaps the third apartment's function is defined starting at t=120, but Alex only rented it from t=96 to t=120, so the rent would be calculated as h(t) for t=120, but that's only one month. That doesn't make sense.Alternatively, perhaps the functions are meant to be applied for the duration Alex was renting each apartment, regardless of the t value. So, for the first apartment, t from 0 to 60, f(t)=800+20t. For the second apartment, t from 60 to 96, g(t)=1200 +10(t -60). For the third apartment, t from 96 to 120, h(t)=1500 +5(t -120). But wait, at t=96, h(t)=1500 +5(96 -120)=1500 -120=1380. That seems odd because the rent would decrease when moving to the third apartment, which might not make sense.Alternatively, maybe the functions are defined as:f(t) =800 +20t for t=0 to 60g(t)=1200 +10(t -60) for t=60 to 96h(t)=1500 +5(t -96) for t=96 to 120But the problem states h(t)=1500 +5(t -120) for t >=120, which is different.This is a bit of a problem. Maybe I need to proceed with the given functions, even if it seems inconsistent, because otherwise, I can't solve the problem.So, let's proceed step by step.First, the first apartment was rented for 5 years, which is 60 months. So, t goes from 0 to 60.The rent for the first apartment is f(t)=800 +20t. So, each month, the rent increases by 20.To find the total rent paid for the first apartment, we need to sum f(t) from t=0 to t=59 (since at t=60, Alex moves to the second apartment).Wait, but actually, at t=60, Alex moves, so the last payment for the first apartment is at t=59. So, the total rent is the sum from t=0 to t=59 of f(t).Similarly, for the second apartment, rented from t=60 to t=95 (since 3 years is 36 months, so t=60 to t=95, inclusive). Then, the rent is g(t)=1200 +10(t -60) for t >=60. So, at t=60, g(60)=1200 +10(0)=1200. At t=61, it's 1210, and so on.So, the total rent for the second apartment is the sum from t=60 to t=95 of g(t).Similarly, the third apartment was rented from t=96 to t=119 (2 years is 24 months, so t=96 to t=119). The rent is h(t)=1500 +5(t -120) for t >=120. Wait, but Alex only rented it until t=119. So, this is a problem because h(t) is only defined for t >=120, but Alex was renting it from t=96 to t=119. So, perhaps the function is supposed to be h(t)=1500 +5(t -96) for t >=96, but the problem says h(t)=1500 +5(t -120) for t >=120. So, maybe it's a typo, and it should be h(t)=1500 +5(t -96) for t >=96.Alternatively, perhaps the rent for the third apartment is flat at 1500 for the first 24 months, but that doesn't fit the function given.Alternatively, maybe the function is correct, and the rent only starts increasing after t=120, but Alex only rented it until t=119, so the rent is flat at 1500 for the entire duration. That could be possible.Wait, let's think. If h(t)=1500 +5(t -120) for t >=120, then for t <120, h(t) is undefined or perhaps constant? If it's undefined, then maybe the rent is 1500 for t <120. But the problem doesn't specify that. It just says h(t)=1500 +5(t -120) for t >=120, so for t <120, h(t) is not defined. Therefore, perhaps the rent for the third apartment is 1500 for the entire duration, since the function isn't defined before t=120.But that seems inconsistent with the first two functions, which are defined for their respective periods.Alternatively, maybe the problem intended for h(t) to be 1500 +5(t -96) for t >=96, but it's written as 1500 +5(t -120). Maybe it's a typo. Since the third apartment was rented for 2 years starting at t=96, the function should be h(t)=1500 +5(t -96) for t >=96.Given that, I think it's safe to assume that it's a typo, and the function should be h(t)=1500 +5(t -96) for t >=96, because otherwise, the rent would only start increasing after t=120, which is after Alex stopped renting it.Therefore, I will proceed with that assumption.So, to summarize:First apartment: t=0 to t=59 (60 months), rent f(t)=800 +20tSecond apartment: t=60 to t=95 (36 months), rent g(t)=1200 +10(t -60)Third apartment: t=96 to t=119 (24 months), rent h(t)=1500 +5(t -96)Now, I need to calculate the total rent paid for each apartment and sum them up.Let's start with the first apartment.First apartment: f(t)=800 +20t, t=0 to 59.This is an arithmetic sequence where the first term a1 = f(0)=800, and the last term a60 = f(59)=800 +20*59=800 +1180=1980.The number of terms is 60.The sum S1 = n*(a1 + a_n)/2 = 60*(800 +1980)/2 = 60*(2780)/2 = 60*1390 = 83,400.Wait, let me calculate that again.a1 =800a60=800 +20*59=800 +1180=1980Sum S1= (number of terms)/2 * (first term + last term) =60/2*(800 +1980)=30*(2780)=83,400.Yes, that's correct.So, total rent for the first apartment is 83,400.Now, the second apartment: g(t)=1200 +10(t -60), t=60 to 95.Let's find the first term and the last term.At t=60: g(60)=1200 +10*(0)=1200At t=95: g(95)=1200 +10*(95 -60)=1200 +10*35=1200 +350=1550Number of terms: from t=60 to t=95 is 36 months.So, sum S2=36/2*(1200 +1550)=18*(2750)=49,500.Wait, let me verify:First term a1=1200Last term a36=1550Sum= (36/2)*(1200 +1550)=18*(2750)=49,500.Yes, correct.So, total rent for the second apartment is 49,500.Now, the third apartment: h(t)=1500 +5(t -96), t=96 to 119.First term at t=96: h(96)=1500 +5*(0)=1500Last term at t=119: h(119)=1500 +5*(119 -96)=1500 +5*23=1500 +115=1615Number of terms: from t=96 to t=119 is 24 months.Sum S3=24/2*(1500 +1615)=12*(3115)=37,380.Wait, let me calculate:1500 +1615=31153115*12=37,380.Yes, correct.So, total rent for the third apartment is 37,380.Now, total rent over 10 years is S1 + S2 + S3=83,400 +49,500 +37,380.Let me add them up:83,400 +49,500=132,900132,900 +37,380=170,280.So, total rent paid over 10 years is 170,280.Wait, let me double-check the calculations.First apartment: 60 months, average rent (800 +1980)/2=1390, total 60*1390=83,400. Correct.Second apartment: 36 months, average rent (1200 +1550)/2=1375, total 36*1375=49,500. Correct.Third apartment: 24 months, average rent (1500 +1615)/2=1557.5, total 24*1557.5=37,380. Correct.Total: 83,400 +49,500=132,900; 132,900 +37,380=170,280.Yes, that seems correct.So, the answer to the first sub-problem is 170,280.Now, moving on to the second sub-problem: calculating the expected cost Alex should budget for moving expenses over the next 10 years, given that the probability of being forced to move in any given year is 0.1, and the cost of moving is approximately 5,000 each time.This is an expected value problem. The expected number of moves over 10 years can be calculated, and then multiplied by the cost per move.The probability of moving in any given year is 0.1, so the expected number of moves per year is 0.1. Over 10 years, the expected number of moves is 10 *0.1=1.Therefore, the expected cost is 1 * 5,000= 5,000.Wait, but actually, the expected number of moves is a Poisson process, but since each year is independent, the expected number is indeed the sum of the expectations each year. So, each year, the expected number of moves is 0.1, so over 10 years, it's 10*0.1=1.Therefore, the expected cost is 1*5,000= 5,000.But wait, another way to think about it is that each year, there's a 10% chance of moving, so the expected number of moves is 10*0.1=1, so the expected cost is 1*5,000= 5,000.Alternatively, we can model it as the expected value:The expected cost E = sum over each year of (probability of moving that year * cost).Since each year is independent, and the probability each year is 0.1, then E=10*(0.1*5000)=10*500=5000.Yes, same result.Therefore, the expected cost Alex should budget is 5,000.Wait, but let me think again. Is the probability of moving in any given year 0.1, meaning that each year, there's a 10% chance of moving once. So, the expected number of moves is 10*0.1=1, so expected cost is 1*5000=5000.Alternatively, if the probability was of moving at least once in 10 years, it would be different, but the problem states "the probability of being forced to move in any given year is 0.1", so it's per year.Therefore, the expected number of moves is 1, so expected cost is 5,000.Yes, that seems correct.So, the answer to the second sub-problem is 5,000.Final Answer1. The total rent paid over the 10-year period is boxed{170280} dollars.2. The expected moving expenses over the next 10 years is boxed{5000} dollars.</think>

🧠:<think>Alright, so I have two problems to solve here. Let me tackle them one by one.Starting with the first problem: It's about calculating the probability of experiencing at least three paranormal events during a haunted history tour in Washington, D.C. The tour visits five haunted sites, each with different probabilities of encountering a paranormal event. These probabilities are 0.2, 0.3, 0.4, 0.25, and 0.35. The events are independent, so I can model this using the binomial probability formula, but wait, actually, each site has a different probability, so it's not a binomial distribution but rather a Poisson binomial distribution. Hmm, that might complicate things a bit.Okay, so the Poisson binomial distribution is used when we have independent trials with different probabilities of success. In this case, each site is a trial, and a "success" would be encountering a paranormal event. We need the probability of getting at least three successes out of five trials.The formula for the Poisson binomial distribution is a bit involved. The probability of exactly k successes is the sum over all combinations of k successes multiplied by the product of their probabilities and the product of the probabilities of failure for the remaining trials. So, for at least three successes, we need to calculate the probabilities for exactly 3, 4, and 5 successes and sum them up.Let me denote the probabilities as p1 = 0.2, p2 = 0.3, p3 = 0.4, p4 = 0.25, p5 = 0.35. The corresponding failure probabilities are q1 = 1 - p1 = 0.8, q2 = 0.7, q3 = 0.6, q4 = 0.75, q5 = 0.65.First, let's compute the probability of exactly 3 events. There are C(5,3) = 10 possible combinations. For each combination, I need to multiply the probabilities of success for the three chosen sites and the probabilities of failure for the remaining two.This seems tedious, but let's try to compute it step by step.1. Combination 1,2,3: p1*p2*p3*q4*q5 = 0.2*0.3*0.4*0.75*0.652. Combination 1,2,4: p1*p2*p4*q3*q5 = 0.2*0.3*0.25*0.6*0.653. Combination 1,2,5: p1*p2*p5*q3*q4 = 0.2*0.3*0.35*0.6*0.754. Combination 1,3,4: p1*p3*p4*q2*q5 = 0.2*0.4*0.25*0.7*0.655. Combination 1,3,5: p1*p3*p5*q2*q4 = 0.2*0.4*0.35*0.7*0.756. Combination 1,4,5: p1*p4*p5*q2*q3 = 0.2*0.25*0.35*0.7*0.67. Combination 2,3,4: p2*p3*p4*q1*q5 = 0.3*0.4*0.25*0.8*0.658. Combination 2,3,5: p2*p3*p5*q1*q4 = 0.3*0.4*0.35*0.8*0.759. Combination 2,4,5: p2*p4*p5*q1*q3 = 0.3*0.25*0.35*0.8*0.610. Combination 3,4,5: p3*p4*p5*q1*q2 = 0.4*0.25*0.35*0.8*0.7Wow, that's a lot. Let me compute each one:1. 0.2*0.3 = 0.06; 0.06*0.4 = 0.024; 0.024*0.75 = 0.018; 0.018*0.65 = 0.01172. 0.2*0.3 = 0.06; 0.06*0.25 = 0.015; 0.015*0.6 = 0.009; 0.009*0.65 = 0.005853. 0.2*0.3 = 0.06; 0.06*0.35 = 0.021; 0.021*0.6 = 0.0126; 0.0126*0.75 = 0.009454. 0.2*0.4 = 0.08; 0.08*0.25 = 0.02; 0.02*0.7 = 0.014; 0.014*0.65 = 0.00915. 0.2*0.4 = 0.08; 0.08*0.35 = 0.028; 0.028*0.7 = 0.0196; 0.0196*0.75 = 0.01476. 0.2*0.25 = 0.05; 0.05*0.35 = 0.0175; 0.0175*0.7 = 0.01225; 0.01225*0.6 = 0.007357. 0.3*0.4 = 0.12; 0.12*0.25 = 0.03; 0.03*0.8 = 0.024; 0.024*0.65 = 0.01568. 0.3*0.4 = 0.12; 0.12*0.35 = 0.042; 0.042*0.8 = 0.0336; 0.0336*0.75 = 0.02529. 0.3*0.25 = 0.075; 0.075*0.35 = 0.02625; 0.02625*0.8 = 0.021; 0.021*0.6 = 0.012610. 0.4*0.25 = 0.1; 0.1*0.35 = 0.035; 0.035*0.8 = 0.028; 0.028*0.7 = 0.0196Now, let's sum all these up:1. 0.01172. 0.005853. 0.009454. 0.00915. 0.01476. 0.007357. 0.01568. 0.02529. 0.012610. 0.0196Adding them step by step:Start with 0.0117 + 0.00585 = 0.017550.01755 + 0.00945 = 0.0270.027 + 0.0091 = 0.03610.0361 + 0.0147 = 0.05080.0508 + 0.00735 = 0.058150.05815 + 0.0156 = 0.073750.07375 + 0.0252 = 0.098950.09895 + 0.0126 = 0.111550.11155 + 0.0196 = 0.13115So, the probability of exactly 3 events is approximately 0.13115.Now, moving on to exactly 4 events. There are C(5,4) = 5 combinations.1. Combination 1,2,3,4: p1*p2*p3*p4*q52. Combination 1,2,3,5: p1*p2*p3*p5*q43. Combination 1,2,4,5: p1*p2*p4*p5*q34. Combination 1,3,4,5: p1*p3*p4*p5*q25. Combination 2,3,4,5: p2*p3*p4*p5*q1Calculating each:1. 0.2*0.3 = 0.06; 0.06*0.4 = 0.024; 0.024*0.25 = 0.006; 0.006*0.65 = 0.00392. 0.2*0.3 = 0.06; 0.06*0.4 = 0.024; 0.024*0.35 = 0.0084; 0.0084*0.75 = 0.00633. 0.2*0.3 = 0.06; 0.06*0.25 = 0.015; 0.015*0.35 = 0.00525; 0.00525*0.6 = 0.003154. 0.2*0.4 = 0.08; 0.08*0.25 = 0.02; 0.02*0.35 = 0.007; 0.007*0.7 = 0.00495. 0.3*0.4 = 0.12; 0.12*0.25 = 0.03; 0.03*0.35 = 0.0105; 0.0105*0.8 = 0.0084Adding these up:1. 0.00392. 0.00633. 0.003154. 0.00495. 0.0084Sum:0.0039 + 0.0063 = 0.01020.0102 + 0.00315 = 0.013350.01335 + 0.0049 = 0.018250.01825 + 0.0084 = 0.02665So, the probability of exactly 4 events is approximately 0.02665.Now, for exactly 5 events. There's only one combination: all five sites have events.Probability = p1*p2*p3*p4*p5 = 0.2*0.3*0.4*0.25*0.35Calculating:0.2*0.3 = 0.060.06*0.4 = 0.0240.024*0.25 = 0.0060.006*0.35 = 0.0021So, the probability of exactly 5 events is 0.0021.Now, adding up the probabilities for exactly 3, 4, and 5 events:0.13115 + 0.02665 + 0.0021 = ?0.13115 + 0.02665 = 0.15780.1578 + 0.0021 = 0.1599So, approximately 0.1599, or 15.99%.Wait, that seems a bit low. Let me double-check my calculations because it's easy to make errors in these multiplications.Looking back at the exactly 3 events:1. 0.01172. 0.005853. 0.009454. 0.00915. 0.01476. 0.007357. 0.01568. 0.02529. 0.012610. 0.0196Adding them again:0.0117 + 0.00585 = 0.01755+0.00945 = 0.027+0.0091 = 0.0361+0.0147 = 0.0508+0.00735 = 0.05815+0.0156 = 0.07375+0.0252 = 0.09895+0.0126 = 0.11155+0.0196 = 0.13115That seems correct.Exactly 4 events:0.0039 + 0.0063 + 0.00315 + 0.0049 + 0.0084 = 0.02665Yes, that's correct.Exactly 5 events: 0.0021So total is 0.13115 + 0.02665 + 0.0021 = 0.1599.So approximately 15.99%, which is roughly 16%.But let me think, is there another way to compute this? Maybe using complementary probability? The probability of at least 3 events is 1 minus the probability of 0, 1, or 2 events.But computing 0,1,2 might be more work, but let's see.Alternatively, maybe using generating functions? The generating function for the Poisson binomial distribution is the product of (1 - pi + pi*x) for each i.So, the generating function G(x) = (1 - 0.2 + 0.2x)(1 - 0.3 + 0.3x)(1 - 0.4 + 0.4x)(1 - 0.25 + 0.25x)(1 - 0.35 + 0.35x)We need the coefficients of x^3, x^4, x^5 in G(x) and sum them.But computing this manually would be time-consuming, but maybe we can compute it step by step.Alternatively, maybe using logarithms or something, but that might not be straightforward.Alternatively, perhaps using inclusion-exclusion, but that might also be complicated.Alternatively, maybe using recursion or dynamic programming, but that's more of a programming approach.Alternatively, perhaps using the fact that the expected number of events is the sum of the probabilities, which is 0.2 + 0.3 + 0.4 + 0.25 + 0.35 = 1.5.But the expected value is 1.5, so the probability of at least 3 events is not too high, which aligns with our previous result of ~16%.Alternatively, maybe using normal approximation, but since n is small (n=5), it's not very accurate.Alternatively, let me check my calculations again for exactly 3 events.Looking at combination 1,2,3: 0.2*0.3*0.4*0.75*0.65.Wait, 0.2*0.3 is 0.06, *0.4 is 0.024, *0.75 is 0.018, *0.65 is 0.0117. Correct.Combination 1,2,4: 0.2*0.3*0.25*0.6*0.65.0.2*0.3=0.06, *0.25=0.015, *0.6=0.009, *0.65=0.00585. Correct.Combination 1,2,5: 0.2*0.3*0.35*0.6*0.75.0.2*0.3=0.06, *0.35=0.021, *0.6=0.0126, *0.75=0.00945. Correct.Combination 1,3,4: 0.2*0.4*0.25*0.7*0.65.0.2*0.4=0.08, *0.25=0.02, *0.7=0.014, *0.65=0.0091. Correct.Combination 1,3,5: 0.2*0.4*0.35*0.7*0.75.0.2*0.4=0.08, *0.35=0.028, *0.7=0.0196, *0.75=0.0147. Correct.Combination 1,4,5: 0.2*0.25*0.35*0.7*0.6.0.2*0.25=0.05, *0.35=0.0175, *0.7=0.01225, *0.6=0.00735. Correct.Combination 2,3,4: 0.3*0.4*0.25*0.8*0.65.0.3*0.4=0.12, *0.25=0.03, *0.8=0.024, *0.65=0.0156. Correct.Combination 2,3,5: 0.3*0.4*0.35*0.8*0.75.0.3*0.4=0.12, *0.35=0.042, *0.8=0.0336, *0.75=0.0252. Correct.Combination 2,4,5: 0.3*0.25*0.35*0.8*0.6.0.3*0.25=0.075, *0.35=0.02625, *0.8=0.021, *0.6=0.0126. Correct.Combination 3,4,5: 0.4*0.25*0.35*0.8*0.7.0.4*0.25=0.1, *0.35=0.035, *0.8=0.028, *0.7=0.0196. Correct.So, all the individual probabilities are correct. So the sum is indeed 0.13115 for exactly 3.Similarly, for exactly 4, the calculations seem correct.So, the total probability is approximately 0.1599, which is about 16%.So, the answer to the first problem is approximately 16%.Now, moving on to the second problem.The tour guide uses a "spooky number" system based on the Fibonacci sequence. The nth haunted site corresponds to the nth Fibonacci number. The total number of visitors is equal to the sum of the first 5 Fibonacci numbers. Then, we need to find the ratio of the total number of visitors to the number of unique Fibonacci numbers less than or equal to this total.First, let's recall the Fibonacci sequence. The Fibonacci sequence starts with F1=1, F2=1, and each subsequent number is the sum of the two preceding ones.So, the first 5 Fibonacci numbers are:F1 = 1F2 = 1F3 = F1 + F2 = 1 + 1 = 2F4 = F2 + F3 = 1 + 2 = 3F5 = F3 + F4 = 2 + 3 = 5So, the first five Fibonacci numbers are 1, 1, 2, 3, 5.The total number of visitors is the sum of these: 1 + 1 + 2 + 3 + 5.Calculating that: 1 + 1 = 2; 2 + 2 = 4; 4 + 3 = 7; 7 + 5 = 12.So, the total number of visitors is 12.Now, we need to determine the number of unique Fibonacci numbers that are less than or equal to 12.First, let's list the Fibonacci numbers up to 12.We already have the first five: 1, 1, 2, 3, 5.Next, F6 = F4 + F5 = 3 + 5 = 8F7 = F5 + F6 = 5 + 8 = 13But 13 is greater than 12, so we stop here.So, the Fibonacci numbers less than or equal to 12 are: 1, 2, 3, 5, 8.But wait, F1 and F2 are both 1, but we need unique numbers. So, the unique Fibonacci numbers ≤12 are: 1, 2, 3, 5, 8.So, that's 5 unique numbers.Therefore, the ratio is total visitors (12) divided by the number of unique Fibonacci numbers (5).So, 12 / 5 = 2.4.But since the problem might expect a fraction, 12/5 is already in simplest terms.So, the ratio is 12:5 or 12/5.Alternatively, if expressed as a decimal, it's 2.4, but probably better to leave it as a fraction.So, the total number of visitors is 12, and the ratio is 12/5.Wait, let me double-check the Fibonacci numbers up to 12.F1=1, F2=1, F3=2, F4=3, F5=5, F6=8, F7=13.Yes, so up to F6=8, which is ≤12, and F7=13 is >12. So, unique Fibonacci numbers ≤12 are 1,2,3,5,8. That's 5 numbers.So, the ratio is 12/5.Therefore, the answers are:1. Approximately 16% probability.2. Total visitors: 12; ratio: 12/5.But let me express the first answer more precisely. Earlier, I got 0.1599, which is approximately 0.16, so 16%.Alternatively, maybe I can compute it more accurately.Wait, 0.1599 is approximately 0.16, which is 16%.Alternatively, maybe I can express it as a fraction.0.1599 is roughly 16/100, which simplifies to 4/25, which is 0.16.But 0.1599 is very close to 0.16, so 16% is a good approximation.Alternatively, maybe I can compute it more precisely.Wait, 0.1599 is approximately 16/94, but that's not helpful.Alternatively, maybe I can leave it as 0.1599, but since the question doesn't specify the form, probably 16% is acceptable.Alternatively, perhaps the exact value is 0.1599, which is approximately 16%.So, summarizing:1. The probability is approximately 16%.2. The total number of visitors is 12, and the ratio is 12/5.Final Answer1. The probability is boxed{0.16}.2. The total number of visitors is boxed{12} and the ratio is boxed{dfrac{12}{5}}.</think>

🧠:<think>Okay, so I have this problem about animal breeding and genetics. It's about a graduate student studying dairy cattle and optimizing genetic diversity. The problem has two parts. Let me try to tackle them one by one.Starting with the first part: We have a population of 200 cattle, 100 males and 100 females. Each has a pair of alleles for a particular gene, with two alleles, A and a. The initial frequencies are 0.6 for A and 0.4 for a. We need to calculate the expected genotype frequencies (AA, Aa, aa) in the first generation using the Hardy-Weinberg principle. Then, derive the inbreeding coefficient (F) for the second generation, assuming random mating.Alright, Hardy-Weinberg principle. I remember that it's a model that predicts genotype frequencies in a population assuming no evolution (no selection, mutation, genetic drift, migration, or non-random mating). The formula is p² + 2pq + q² = 1, where p is the frequency of allele A, and q is the frequency of allele a.Given p = 0.6 and q = 0.4, the genotype frequencies should be:- AA: p² = 0.6² = 0.36- Aa: 2pq = 2*0.6*0.4 = 0.48- aa: q² = 0.4² = 0.16So, in the first generation, the expected genotype frequencies are 36% AA, 48% Aa, and 16% aa.Now, moving on to the inbreeding coefficient (F) for the second generation. Hmm, inbreeding coefficient is a measure of the likelihood of an individual inheriting two identical alleles from its parents. It's often used to quantify the level of inbreeding in a population.Wait, but how do we calculate F for the second generation? I think inbreeding can occur due to non-random mating, like inbreeding within a population. But in this case, the problem says random mating. So, if it's random mating, the inbreeding coefficient should be zero, right? Because in random mating, there's no increased likelihood of individuals mating with relatives.But wait, maybe I'm misunderstanding. The inbreeding coefficient can also be calculated based on the change in genotype frequencies over generations. Let me recall the formula for inbreeding.The inbreeding coefficient F is given by F = (Heterozygosity under random mating - Observed heterozygosity) / (Heterozygosity under random mating). Or is it the other way around? Wait, no, I think it's F = (Observed heterozygosity - Expected heterozygosity) / Expected heterozygosity. Or maybe it's the other way. Let me check.Wait, actually, the formula is F = (2pq - H) / (2pq), where H is the observed heterozygosity. But in the first generation, if we're using Hardy-Weinberg, the heterozygosity is 2pq, which is 0.48. If in the second generation, due to some reason, the heterozygosity changes, then we can compute F.But the problem says to derive F for the second generation, assuming random mating. So, if mating is random, the genotype frequencies should remain the same as per Hardy-Weinberg. So, in that case, the heterozygosity should still be 0.48, so F would be zero.Wait, but maybe I'm overcomplicating. Alternatively, the inbreeding coefficient can be calculated using the formula F = 1 - (sum of genotype frequencies squared). Wait, no, that's the formula for something else.Wait, another approach: The inbreeding coefficient can also be calculated based on the probability of an individual inheriting the same allele from both parents. So, for a population, F is the probability that the two alleles in an individual are identical by descent.In a randomly mating population, F is zero. So, if the population is in Hardy-Weinberg equilibrium, F is zero. So, if we have random mating, the inbreeding coefficient remains zero.But wait, the problem is asking for the inbreeding coefficient for the second generation. So, perhaps they are considering that in the first generation, the population is in HWE, but in the second generation, due to some reason, inbreeding occurs.But the problem says "assuming random mating within the population." So, if it's random mating, then F remains zero. So, maybe the inbreeding coefficient for the second generation is zero.Alternatively, perhaps the question is about the change in allele frequencies due to inbreeding. Wait, but inbreeding doesn't change allele frequencies, it just changes genotype frequencies. So, if allele frequencies remain the same, but genotype frequencies change, then F can be calculated.Wait, let me think again. The inbreeding coefficient is calculated as F = (Observed heterozygosity - Expected heterozygosity) / (Expected heterozygosity). So, if in the second generation, the observed heterozygosity is less than expected, F would be positive.But in this case, since mating is random, the observed heterozygosity should equal the expected heterozygosity, so F = 0.Alternatively, perhaps the question is referring to the inbreeding coefficient of an individual, but in a randomly mating population, the average inbreeding coefficient is zero.So, I think the inbreeding coefficient F for the second generation is zero.Wait, but maybe I'm missing something. Let me check the formula for inbreeding coefficient in terms of genotype frequencies.The inbreeding coefficient can be calculated as:F = (p² + q² - (p² + q²)') / (2pq)Wait, no, that doesn't seem right.Alternatively, the formula for F is:F = (Observed frequency of homozygotes - Expected frequency of homozygotes) / (Expected frequency of homozygotes)But no, that's not quite right either.Wait, perhaps it's better to use the formula:F = 1 - (sum of genotype frequencies squared). Wait, no, that's the formula for something else.Wait, let me recall: The inbreeding coefficient is the probability that two alleles in an individual are identical by descent. In a randomly mating population, this is zero. So, if the population is in HWE, F is zero.Therefore, if the population is mating randomly, the inbreeding coefficient remains zero.So, perhaps the answer is F = 0 for the second generation.But let me think again. Maybe the question is about the change in allele frequencies over generations due to inbreeding. But inbreeding doesn't change allele frequencies, only genotype frequencies.So, if allele frequencies remain the same, and mating is random, then genotype frequencies remain the same, so F remains zero.Therefore, I think the inbreeding coefficient for the second generation is zero.Okay, moving on to the second part: The student introduces a migration model where 10 new cattle are introduced every generation, with allelic frequencies of 0.7 for A and 0.3 for a. The population size remains constant at 200. We need to derive the recurrence relation for the allelic frequencies in subsequent generations and predict the allelic frequencies after 5 generations.So, this is a migration model where each generation, 10 new individuals are added, and 10 are removed to keep the population size constant. The new individuals have allele frequencies p' = 0.7 and q' = 0.3.We need to model the change in allele frequency over generations due to migration.Let me recall that in migration models, the change in allele frequency can be modeled using the formula:p_{t+1} = (N * p_t + m * p') / (N + m)Where N is the population size, m is the number of migrants, and p' is the allele frequency in the migrant population.But in this case, the population size remains constant, so the number of migrants is 10, and the number of individuals replaced is 10. So, it's more like a constant population size with migration.Wait, actually, the formula for allele frequency change with migration is:p_{t+1} = ( (N - m) * p_t + m * p' ) / NWhere N is the population size, m is the number of migrants, and p' is the allele frequency in the migrant population.In this case, N = 200, m = 10, p' = 0.7.So, plugging in the numbers:p_{t+1} = ( (200 - 10) * p_t + 10 * 0.7 ) / 200Simplify:p_{t+1} = (190 * p_t + 7) / 200So, that's the recurrence relation.Now, we need to predict the allelic frequencies after 5 generations.Given that the initial p0 = 0.6.Let me compute p1, p2, p3, p4, p5.First, p0 = 0.6Compute p1:p1 = (190 * 0.6 + 7) / 200Calculate numerator: 190 * 0.6 = 114; 114 + 7 = 121So, p1 = 121 / 200 = 0.605p1 = 0.605Now, p2:p2 = (190 * 0.605 + 7) / 200Calculate 190 * 0.605: 190 * 0.6 = 114; 190 * 0.005 = 0.95; total = 114 + 0.95 = 114.95Add 7: 114.95 + 7 = 121.95p2 = 121.95 / 200 = 0.60975p2 ≈ 0.60975p3:p3 = (190 * 0.60975 + 7) / 200Calculate 190 * 0.60975:First, 190 * 0.6 = 114190 * 0.00975 = 190 * 0.01 = 1.9; subtract 190 * 0.00025 = 0.0475; so 1.9 - 0.0475 = 1.8525So, total 114 + 1.8525 = 115.8525Add 7: 115.8525 + 7 = 122.8525p3 = 122.8525 / 200 = 0.6142625p3 ≈ 0.61426p4:p4 = (190 * 0.61426 + 7) / 200Calculate 190 * 0.61426:0.6 * 190 = 1140.01426 * 190 ≈ 2.7094So, total ≈ 114 + 2.7094 ≈ 116.7094Add 7: 116.7094 + 7 = 123.7094p4 = 123.7094 / 200 ≈ 0.618547p4 ≈ 0.61855p5:p5 = (190 * 0.61855 + 7) / 200Calculate 190 * 0.61855:0.6 * 190 = 1140.01855 * 190 ≈ 3.5245Total ≈ 114 + 3.5245 ≈ 117.5245Add 7: 117.5245 + 7 = 124.5245p5 = 124.5245 / 200 ≈ 0.6226225So, p5 ≈ 0.6226So, after 5 generations, the allele frequency of A is approximately 0.6226, and a would be 1 - 0.6226 = 0.3774.Alternatively, we can write the recurrence relation as:p_{t+1} = (190/200) * p_t + (10/200)*0.7Simplify:p_{t+1} = 0.95 * p_t + 0.035This is a linear recurrence relation, which can be solved to find a closed-form expression.The general solution for such a recurrence is:p_t = (p0 - p_eq) * (0.95)^t + p_eqWhere p_eq is the equilibrium frequency when p_{t+1} = p_t = p_eq.To find p_eq:p_eq = 0.95 * p_eq + 0.035p_eq - 0.95 p_eq = 0.0350.05 p_eq = 0.035p_eq = 0.035 / 0.05 = 0.7So, the equilibrium frequency is 0.7, which makes sense because the migrants have p' = 0.7, and over time, the population's allele frequency will approach that.Therefore, the closed-form solution is:p_t = (0.6 - 0.7) * (0.95)^t + 0.7p_t = (-0.1) * (0.95)^t + 0.7So, after t generations, p_t = 0.7 - 0.1*(0.95)^tTherefore, after 5 generations:p5 = 0.7 - 0.1*(0.95)^5Calculate (0.95)^5:0.95^1 = 0.950.95^2 = 0.90250.95^3 ≈ 0.8573750.95^4 ≈ 0.814506250.95^5 ≈ 0.7737809375So, p5 ≈ 0.7 - 0.1*0.7737809375 ≈ 0.7 - 0.07737809375 ≈ 0.62262190625Which matches our earlier calculation of approximately 0.6226.So, the allelic frequencies after 5 generations are approximately p ≈ 0.6226 and q ≈ 0.3774.Therefore, summarizing:1. The expected genotype frequencies in the first generation are AA: 0.36, Aa: 0.48, aa: 0.16. The inbreeding coefficient for the second generation is 0.2. The recurrence relation is p_{t+1} = 0.95 p_t + 0.035, and after 5 generations, the allele frequency of A is approximately 0.6226.Wait, but let me double-check the inbreeding coefficient part. If the population is in HWE, F is zero. So, if mating is random, F remains zero. So, yes, F = 0 for the second generation.Alternatively, if the question is considering that the population is initially in HWE, and then in the second generation, due to some reason, inbreeding occurs, but the problem states that mating is random, so F remains zero.Therefore, I think the inbreeding coefficient is zero.So, to recap:1. Genotype frequencies: AA=0.36, Aa=0.48, aa=0.16. Inbreeding coefficient F=0.2. Recurrence relation: p_{t+1} = 0.95 p_t + 0.035. After 5 generations, p≈0.6226.I think that's it.</think>