🧠:<think>Okay, so I have this problem where a bestselling author hides a quadratic polynomial in the coordinates of three locations in her book. The coordinates are given as Location A (2,14), Location B (5,35), and Location C (7,98). The task is to find the quadratic polynomial P(x) = ax² + bx + c. The roots of this polynomial are the x-coordinates of Location A and Location B, which are 2 and 5. Also, when we plug in x₃ = 7 into P(x), it should equal y₃ = 98. Alright, let me start by recalling that if a quadratic polynomial has roots at x = 2 and x = 5, it can be written in its factored form as P(x) = a(x - 2)(x - 5). That makes sense because when x is 2 or 5, the polynomial equals zero, which are the roots. So, expanding this factored form should give me the standard quadratic form. Let me do that step by step. First, multiply (x - 2)(x - 5):(x - 2)(x - 5) = x² - 5x - 2x + 10 = x² - 7x + 10.So, P(x) = a(x² - 7x + 10). Now, I need to find the value of 'a'. To do that, I can use the third point, which is Location C (7,98). That means when x = 7, P(x) = 98. Let me plug x = 7 into the polynomial:P(7) = a(7² - 7*7 + 10) = a(49 - 49 + 10) = a(10).And we know that P(7) = 98, so:a(10) = 98Therefore, a = 98 / 10 = 9.8.Hmm, 9.8 is a decimal. I wonder if that's correct. Let me double-check my calculations.Wait, 7 squared is 49, 7 times 7 is 49, so 49 - 49 is 0, plus 10 is 10. So, yeah, 10a = 98, so a = 9.8. That seems right. But usually, in these problems, the coefficients are integers. Maybe I made a mistake in the factored form? Let me check again.The roots are 2 and 5, so factored form is correct: (x - 2)(x - 5). Expanding that gives x² - 7x + 10. So, that's correct. Alternatively, maybe I should use another point to find 'a'. Wait, the problem says that the roots are the x-coordinates of Location A and Location B, but it doesn't say that the polynomial passes through those points. Wait, hold on. Let me re-read the problem."The roots of this polynomial are designed to be the x-coordinates of Location A and Location B, and P(x₃) equals the y-coordinate of Location C."Oh, so P(x) has roots at x = 2 and x = 5, so it's zero at those x-values, but the y-coordinates of A and B are not necessarily zero. So, the polynomial is not passing through (2,14) or (5,35), but only has roots at x=2 and x=5, and passes through (7,98). So, that means my initial approach was correct. So, P(x) = a(x - 2)(x - 5). Then, plugging in x=7, P(7)=98, so a=9.8. But 9.8 is 49/5, which is 9.8. Maybe the author wanted a decimal, but perhaps I should write it as a fraction. 49/5 is 9 and 4/5, which is 9.8. Alternatively, maybe I can represent it as a fraction. So, 49/5. Let me write that down.So, P(x) = (49/5)(x² - 7x + 10). Let me expand that to get it into standard form:P(x) = (49/5)x² - (49/5)*7x + (49/5)*10.Calculating each term:First term: (49/5)x².Second term: (49/5)*7 = 343/5, so -343/5 x.Third term: (49/5)*10 = 490/5 = 98.So, P(x) = (49/5)x² - (343/5)x + 98.Hmm, that seems a bit messy with fractions, but it's correct. Alternatively, I can write it as:P(x) = 9.8x² - 68.6x + 98.But since the problem didn't specify the form, either is acceptable, but maybe fractions are better since they are exact.Wait, let me check if this polynomial actually gives the correct value at x=7.Compute P(7):(49/5)(7²) - (343/5)(7) + 98.7² is 49, so (49/5)*49 = (49*49)/5 = 2401/5.Then, (343/5)*7 = (343*7)/5 = 2401/5.So, P(7) = 2401/5 - 2401/5 + 98 = 0 + 98 = 98. Perfect, that's correct.So, I think that's the quadratic polynomial. Alternatively, maybe I can write it in a different way. Let me see if I can factor out something. But 49/5 is a common factor in the first two terms, but the last term is 98, which is 49*2. Maybe that's why.Alternatively, maybe I can write the polynomial as P(x) = (49/5)(x² - 7x + 10). Since x² -7x +10 is the factored form.But yeah, that's correct.So, moving on to the second part. The author encodes a linear transformation T which maps the coordinates of Location B to new coordinates (u, v). The transformation is given by the matrix equation:[ u ] [ 3 -1 ] [x₂][ v ] = [ 2 4 ] [y₂]Given that Location B is (5,35), so x₂=5, y₂=35.So, I need to compute u and v by multiplying the matrix with the vector [5; 35].Let me write that out:u = 3*5 + (-1)*35v = 2*5 + 4*35Compute u:3*5 = 15-1*35 = -35So, u = 15 - 35 = -20Compute v:2*5 = 104*35 = 140So, v = 10 + 140 = 150Therefore, the new coordinates after transformation are (-20, 150).Wait, that seems straightforward. Let me double-check my calculations.For u: 3*5 is 15, minus 35 is -20. Correct.For v: 2*5 is 10, plus 4*35 is 140, so 10 + 140 is 150. Correct.So, the new coordinates are (-20, 150).Wait, just to make sure, is the matrix multiplication correct? The matrix is 2x2, the vector is 2x1, so the result is 2x1.First row: 3*5 + (-1)*35 = 15 -35 = -20Second row: 2*5 + 4*35 = 10 + 140 = 150Yes, that's correct.So, summarizing:1. The quadratic polynomial is P(x) = (49/5)x² - (343/5)x + 98, or in factored form, P(x) = (49/5)(x - 2)(x - 5).2. The new coordinates after transformation are (-20, 150).I think that's all. Let me just check if I can represent the polynomial differently. Maybe factor out 49/5:P(x) = (49/5)(x² - 7x + 10). That's a neat form.Alternatively, if I want to write it as P(x) = ax² + bx + c with fractions:a = 49/5, b = -343/5, c = 98.Alternatively, as decimals:a = 9.8, b = -68.6, c = 98.But since the problem didn't specify, either is fine, but fractions are exact, so maybe better to use fractions.So, I think I'm confident with these answers.Final Answer1. The quadratic polynomial is boxed{frac{49}{5}x^2 - frac{343}{5}x + 98}.2. The new coordinates of Location B after the transformation are boxed{(-20, 150)}.</think>

🧠:<think>Okay, so I have this problem about a fantasy author who is helping new writers through a mentorship program. The problem has two parts, and I need to tackle both. Let me start with the first part.Problem 1: Expressing N in terms of a and rThe author has written N books, each with a different number of pages. The number of pages follows a geometric sequence. The first book has 'a' pages, and the common ratio is 'r'. The total number of pages in all N books is 4096. I need to express N in terms of a and r and then show that a and r must satisfy a specific relationship when N is 5.Alright, so let's recall what a geometric sequence is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio (r). So, the number of pages in each book would be: a, ar, ar², ar³, ..., ar^{N-1}.The total number of pages is the sum of this geometric series. The formula for the sum of the first N terms of a geometric series is:S_N = a * (1 - r^N) / (1 - r) when r ≠ 1.Given that the total number of pages is 4096, we have:a * (1 - r^N) / (1 - r) = 4096.So, that's our equation. We need to express N in terms of a and r. Hmm, but solving for N in this equation might not be straightforward because N is in the exponent. Let me write the equation again:a * (1 - r^N) / (1 - r) = 4096.I can rearrange this equation to solve for (1 - r^N):1 - r^N = (4096 * (1 - r)) / a.Then,r^N = 1 - (4096 * (1 - r)) / a.Hmm, that seems a bit messy. Maybe it's better to write it as:r^N = 1 - (4096/a)(1 - r).But I'm not sure if that helps. Alternatively, maybe we can express N as:N = log_r [1 - (4096/a)(1 - r)].But logarithms with variable bases can be tricky. Maybe it's not necessary to express N explicitly in terms of a and r unless we have more information.Wait, the second part says to show that a and r must satisfy a specific relationship given N = 5. So perhaps when N is 5, we can find a relationship between a and r.So, let's substitute N = 5 into the equation:a * (1 - r^5) / (1 - r) = 4096.So,(1 - r^5)/(1 - r) = 4096 / a.But (1 - r^5)/(1 - r) is a geometric series sum formula. Let me compute that:(1 - r^5)/(1 - r) = 1 + r + r² + r³ + r⁴.So, 1 + r + r² + r³ + r⁴ = 4096 / a.Therefore, a = 4096 / (1 + r + r² + r³ + r⁴).So, that's the relationship between a and r when N = 5. So, a is equal to 4096 divided by the sum of the first five terms of the geometric series with ratio r.Alternatively, if we denote S = 1 + r + r² + r³ + r⁴, then a = 4096 / S.So, that's the specific relationship.But the problem says to express N in terms of a and r. So, in the first part, N is expressed as:N = log_r [1 - (4096/a)(1 - r)].But that seems complicated. Maybe another approach is better.Wait, perhaps we can write it as:(1 - r^N)/(1 - r) = 4096 / a.So,(1 - r^N) = (4096 / a)(1 - r).Then,r^N = 1 - (4096 / a)(1 - r).So,r^N = 1 - (4096 / a) + (4096 / a) r.But I don't know if that helps. Maybe it's better to leave it as:N = log_r [1 - (4096 / a)(1 - r)].But logarithms with variable bases are not very elegant. Maybe it's acceptable.Alternatively, if we take natural logarithms on both sides:ln(r^N) = ln[1 - (4096 / a)(1 - r)].So,N ln r = ln[1 - (4096 / a)(1 - r)].Therefore,N = [ln(1 - (4096 / a)(1 - r))] / ln r.But that's still a complicated expression. Maybe it's not necessary to express N in terms of a and r unless we have specific values.Wait, maybe the first part is just to write the equation, not necessarily to solve for N. Let me re-read the problem."Express N in terms of a and r, and show that a and r must satisfy a specific relationship given N = 5."So, perhaps the first part is just to write the equation:a*(1 - r^N)/(1 - r) = 4096.So, that's the expression relating N, a, and r.Then, for the second part, when N = 5, we can write:a*(1 - r^5)/(1 - r) = 4096.Which simplifies to:a = 4096 / [(1 - r^5)/(1 - r)] = 4096 / (1 + r + r² + r³ + r⁴).So, that's the specific relationship. So, a is equal to 4096 divided by the sum of the first five terms of the geometric series with ratio r.Therefore, for N = 5, a and r must satisfy a = 4096 / (1 + r + r² + r³ + r⁴).So, that's the specific relationship.I think that's the answer for the first part.Problem 2: Fibonacci sequence for workshopsNow, moving on to the second part. The author holds a series of workshops. Each workshop has a number of experienced authors (E) and novice writers (W). Both E and W follow a Fibonacci sequence. Specifically, E₁ = 1, E₂ = 1, and similarly W₁ = 1, W₂ = 1. For the k-th workshop, the total participants Pₖ = Eₖ + Wₖ. The author plans to hold 10 workshops, and we need to find the total number of participants across all workshops and determine if it is a Fibonacci number.Alright, so let's break this down.First, both E and W follow their own Fibonacci sequences, starting with E₁ = 1, E₂ = 1, and W₁ = 1, W₂ = 1. So, each sequence is the standard Fibonacci sequence starting from 1, 1.Therefore, Eₖ and Wₖ are both Fibonacci numbers. Specifically, Eₖ = Fib(k) and Wₖ = Fib(k), where Fib(k) is the k-th Fibonacci number.Therefore, Pₖ = Eₖ + Wₖ = Fib(k) + Fib(k) = 2*Fib(k).So, each workshop's participants are twice the k-th Fibonacci number.Now, we need to find the total number of participants across all 10 workshops. That is, sum_{k=1 to 10} Pₖ = sum_{k=1 to 10} 2*Fib(k) = 2*sum_{k=1 to 10} Fib(k).So, first, let's compute sum_{k=1 to 10} Fib(k). There is a known formula for the sum of the first n Fibonacci numbers. The sum of the first n Fibonacci numbers is Fib(n + 2) - 1.Let me verify that.Yes, the sum from k=1 to n of Fib(k) = Fib(n + 2) - 1.So, for n = 10, sum_{k=1 to 10} Fib(k) = Fib(12) - 1.What's Fib(12)?Let me recall the Fibonacci sequence:Fib(1) = 1Fib(2) = 1Fib(3) = 2Fib(4) = 3Fib(5) = 5Fib(6) = 8Fib(7) = 13Fib(8) = 21Fib(9) = 34Fib(10) = 55Fib(11) = 89Fib(12) = 144So, Fib(12) = 144.Therefore, sum_{k=1 to 10} Fib(k) = 144 - 1 = 143.Therefore, the total participants are 2 * 143 = 286.Now, we need to determine if 286 is a Fibonacci number.Looking back at the Fibonacci sequence:Fib(1) = 1Fib(2) = 1Fib(3) = 2Fib(4) = 3Fib(5) = 5Fib(6) = 8Fib(7) = 13Fib(8) = 21Fib(9) = 34Fib(10) = 55Fib(11) = 89Fib(12) = 144Fib(13) = 233Fib(14) = 377Fib(15) = 610So, 286 is between Fib(13) = 233 and Fib(14) = 377. Since 286 is not equal to any Fibonacci number in the sequence, it is not a Fibonacci number.Wait, but let me check if I missed something. Maybe I miscalculated the sum.Wait, let me compute the sum manually to confirm.Compute sum_{k=1 to 10} Fib(k):Fib(1) = 1Fib(2) = 1Fib(3) = 2Fib(4) = 3Fib(5) = 5Fib(6) = 8Fib(7) = 13Fib(8) = 21Fib(9) = 34Fib(10) = 55Adding them up:1 + 1 = 22 + 2 = 44 + 3 = 77 + 5 = 1212 + 8 = 2020 + 13 = 3333 + 21 = 5454 + 34 = 8888 + 55 = 143.Yes, that's correct. So, the sum is 143, times 2 is 286.Looking at the Fibonacci numbers, 286 is not in the list. Therefore, the total number of participants is 286, which is not a Fibonacci number.Alternatively, maybe I should check if 286 is a Fibonacci number by using the Fibonacci formula or some property.One way to check if a number is a Fibonacci number is to see if 5n² + 4 or 5n² - 4 is a perfect square.So, let's compute 5*(286)^2 + 4 and 5*(286)^2 - 4.First, 286 squared:286 * 286. Let's compute that.286 * 200 = 57,200286 * 80 = 22,880286 * 6 = 1,716Adding them up: 57,200 + 22,880 = 80,080; 80,080 + 1,716 = 81,796.So, 286² = 81,796.Then, 5*81,796 = 408,980.So, 5n² + 4 = 408,980 + 4 = 408,984.Is 408,984 a perfect square?Let me see. Let's compute sqrt(408,984).Well, 640² = 409,600, which is higher than 408,984.639² = (640 - 1)² = 640² - 2*640 + 1 = 409,600 - 1,280 + 1 = 408,321.So, 639² = 408,321.408,321 vs. 408,984. The difference is 408,984 - 408,321 = 663.So, 639² = 408,321640² = 409,600So, 408,984 is between these two, not a perfect square.Similarly, 5n² - 4 = 408,980 - 4 = 408,976.Check if 408,976 is a perfect square.Again, 639² = 408,321640² = 409,600So, 408,976 is between them. Let me see:Compute 639.5²: approximately (639 + 0.5)² = 639² + 2*639*0.5 + 0.25 = 408,321 + 639 + 0.25 = 408,960.25.So, 639.5² ≈ 408,960.25.But 408,976 is higher than that.Compute 639.7²:= (639 + 0.7)² = 639² + 2*639*0.7 + 0.49 = 408,321 + 894.6 + 0.49 ≈ 409,216.09.Wait, that's not right. Wait, 2*639*0.7 = 2*639=1278; 1278*0.7=894.6.So, 639² + 894.6 + 0.49 = 408,321 + 894.6 = 409,215.6 + 0.49 ≈ 409,216.09.But 408,976 is less than that. Wait, perhaps I made a miscalculation.Wait, 639.5² is approximately 408,960.25, as above.So, 408,976 - 408,960.25 = 15.75.So, 639.5 + x squared is 408,976.Approximately, x ≈ 15.75 / (2*639.5) ≈ 15.75 / 1279 ≈ 0.0123.So, sqrt(408,976) ≈ 639.5 + 0.0123 ≈ 639.5123.So, it's not an integer. Therefore, 408,976 is not a perfect square.Therefore, 286 is not a Fibonacci number.Hence, the total number of participants is 286, which is not a Fibonacci number.So, summarizing:1. For the first part, when N = 5, a and r must satisfy a = 4096 / (1 + r + r² + r³ + r⁴).2. For the second part, the total number of participants is 286, which is not a Fibonacci number.Final Answer1. The relationship is ( a = frac{4096}{1 + r + r^2 + r^3 + r^4} ). So, ( boxed{a = frac{4096}{1 + r + r^2 + r^3 + r^4}} ).2. The total number of participants is ( boxed{286} ), which is not a Fibonacci number.</think>

🧠:<think>Alright, so I've got this problem about African American actors in leading roles on TV from 1950 to 1970. It's split into two parts, each covering a decade. Let me try to tackle them one by one.Starting with part 1: The period from 1950 to 1960. They give me a differential equation for the number of African American actors, f(t), where t is years since 1950. The equation is df/dt = k*(10 - f(t)). Hmm, that looks like a logistic growth model or maybe exponential decay? Let me think.So, the differential equation is df/dt = k*(10 - f(t)). This is a first-order linear ordinary differential equation. I remember that these can be solved using integrating factors or recognizing them as separable equations. Let me try to separate variables.Rewriting the equation: df/dt = k*(10 - f). So, I can write this as df/(10 - f) = k*dt. Integrating both sides should give me the solution.The integral of df/(10 - f) is -ln|10 - f| + C, right? And the integral of k*dt is k*t + C. So putting it together:- ln|10 - f| = k*t + CExponentiating both sides to get rid of the natural log:|10 - f| = e^{-k*t - C} = e^{-C} * e^{-k*t}Let me denote e^{-C} as another constant, say, A. So:10 - f = A*e^{-k*t}Therefore, f(t) = 10 - A*e^{-k*t}Now, applying the initial condition f(0) = 2. So when t=0:f(0) = 10 - A*e^{0} = 10 - A = 2So, 10 - A = 2 => A = 8Thus, the function becomes f(t) = 10 - 8*e^{-k*t}Now, we also know that f(10) = 8. So plugging t=10 into the equation:8 = 10 - 8*e^{-10*k}Let me solve for k.Subtract 10 from both sides: 8 - 10 = -8*e^{-10*k} => -2 = -8*e^{-10*k}Divide both sides by -8: (-2)/(-8) = e^{-10*k} => 1/4 = e^{-10*k}Take natural log of both sides: ln(1/4) = -10*kSo, ln(1) - ln(4) = -10*k => 0 - ln(4) = -10*k => -ln(4) = -10*kMultiply both sides by -1: ln(4) = 10*kTherefore, k = ln(4)/10Calculating ln(4): ln(4) is approximately 1.386, so k ≈ 0.1386 per year.But since they didn't specify to approximate, I should keep it exact. So k = (ln 4)/10.So, summarizing part 1: k = (ln 4)/10 and f(t) = 10 - 8*e^{-(ln 4)/10 * t}I can also write e^{(ln 4)/10} as 4^{1/10}, so f(t) = 10 - 8*(4^{-t/10}) or 10 - 8*(2^{-2t/10}) = 10 - 8*(2^{-t/5})But maybe it's better to leave it in exponential form with base e.Moving on to part 2: The period from 1960 to 1970. They model the number of actors with a quadratic function g(t) = a*t^2 + b*t + c, where t is years since 1960. They give three points: in 1960 (t=0), g(0)=8; in 1965 (t=5), g(5)=20; and in 1970 (t=10), g(10)=32.So, we have three equations:1. When t=0: g(0) = a*0 + b*0 + c = c = 8 => c=82. When t=5: g(5) = a*(5)^2 + b*(5) + c = 25a + 5b + 8 = 203. When t=10: g(10) = a*(10)^2 + b*(10) + c = 100a + 10b + 8 = 32So, now we can set up the equations:From equation 2: 25a + 5b + 8 = 20 => 25a + 5b = 12From equation 3: 100a + 10b + 8 = 32 => 100a + 10b = 24Now, let's simplify these equations.Equation 2: 25a + 5b = 12. Let's divide both sides by 5: 5a + b = 12/5 = 2.4Equation 3: 100a + 10b = 24. Let's divide both sides by 10: 10a + b = 2.4Wait, so now we have:Equation 2 simplified: 5a + b = 2.4Equation 3 simplified: 10a + b = 2.4Hmm, subtract equation 2 from equation 3:(10a + b) - (5a + b) = 2.4 - 2.4 => 5a = 0 => a=0Wait, that can't be right because if a=0, then the quadratic becomes linear. Let me check my calculations.Wait, equation 2: 25a + 5b = 12Equation 3: 100a + 10b = 24If I multiply equation 2 by 2: 50a + 10b = 24Then subtract equation 3: (50a + 10b) - (100a + 10b) = 24 - 24 => -50a = 0 => a=0So, a=0. Then plugging back into equation 2: 25*0 + 5b = 12 => 5b=12 => b=12/5=2.4So, a=0, b=2.4, c=8Therefore, the quadratic function is g(t)=0*t^2 + 2.4*t +8=2.4t +8Wait, but that's a linear function, not quadratic. So, the model they gave is quadratic, but with a=0, it's linear. That seems odd. Maybe I made a mistake.Wait, let me check the equations again.Given:g(0)=8: correct, so c=8.g(5)=20: 25a +5b +8=20 =>25a +5b=12g(10)=32:100a +10b +8=32 =>100a +10b=24So, 25a +5b=12 and 100a +10b=24If I multiply the first equation by 2: 50a +10b=24Then subtract the second equation: (50a +10b) - (100a +10b)=24 -24 => -50a=0 => a=0So, yes, a=0. Then b=12/5=2.4So, the quadratic reduces to linear. Maybe the data points lie on a straight line, so the quadratic term isn't necessary. That's possible.So, the function is g(t)=2.4t +8But let me verify with the points:At t=0: 2.4*0 +8=8: correct.At t=5:2.4*5 +8=12 +8=20: correct.At t=10:2.4*10 +8=24 +8=32: correct.So, even though it's supposed to be quadratic, the data fits a linear model. So, a=0, b=2.4, c=8.So, in conclusion, part 2: a=0, b=2.4, c=8.But let me think again: is it possible that the quadratic model is intended, but the data is linear? Maybe the question expects a quadratic, so perhaps I made a mistake in interpreting the points.Wait, let me double-check the points:1960 (t=0): 81965 (t=5):201970 (t=10):32So, the differences:From t=0 to t=5: increase of 12 over 5 years.From t=5 to t=10: increase of 12 over 5 years.So, the rate is constant, hence linear. So, the quadratic model with a=0 is correct.Therefore, the coefficients are a=0, b=2.4, c=8.I think that's it.Final Answer1. The value of (k) is (boxed{dfrac{ln 4}{10}}) and the function is (f(t) = 10 - 8e^{-frac{ln 4}{10} t}).2. The coefficients are (a = boxed{0}), (b = boxed{dfrac{12}{5}}), and (c = boxed{8}).</think>

🧠:<think>Alright, so I have this problem about the St. Louis Blues hockey team, and I need to figure out two things. First, I need to find the maximum likelihood estimate of their probability of winning a game, given that they won 46 out of 82 games. Second, I need to calculate the probability that they scored exactly 5 goals in a randomly chosen game, and also find the variance of the number of goals they scored per game. Hmm, okay, let me take this step by step.Starting with the first part: maximum likelihood estimate of ( p ), the probability of winning a game. I remember that maximum likelihood estimation is a method used to estimate the parameters of a statistical model. In this case, the model is a binomial distribution because each game is an independent event with two possible outcomes: win or loss. The number of trials is 82 games, and the number of successes is 46 wins.The likelihood function for a binomial distribution is given by:[L(p) = binom{n}{k} p^k (1-p)^{n-k}]Where ( n ) is the number of trials, ( k ) is the number of successes, and ( p ) is the probability of success. To find the maximum likelihood estimate, we need to maximize this function with respect to ( p ).I also recall that the maximum likelihood estimate for ( p ) in a binomial distribution is simply the sample proportion of successes. So, in this case, it would be the number of wins divided by the total number of games. That makes sense because intuitively, the best estimate for the probability of winning is just the proportion of games they actually won.So, plugging in the numbers:Number of wins, ( k = 46 )Total number of games, ( n = 82 )Therefore, the maximum likelihood estimate ( hat{p} ) is:[hat{p} = frac{k}{n} = frac{46}{82}]Let me compute that. 46 divided by 82. Well, 46 is half of 92, which is 82 plus 10. Hmm, maybe I should just do the division.46 divided by 82. Let me see, 82 goes into 46 zero times. Add a decimal point, 82 goes into 460 five times because 5*82 is 410. Subtract 410 from 460, we get 50. Bring down the next zero: 500. 82 goes into 500 six times because 6*82 is 492. Subtract 492 from 500, we get 8. Bring down another zero: 80. 82 goes into 80 zero times. So, we have 0.560... approximately.Wait, let me check that again. 46 divided by 82. Maybe I can simplify the fraction first. Both 46 and 82 are divisible by 2. So, 46 divided by 2 is 23, and 82 divided by 2 is 41. So, it's 23/41. Let me compute 23 divided by 41.41 goes into 23 zero times. 41 goes into 230 five times (5*41=205). Subtract 205 from 230, we get 25. Bring down a zero: 250. 41 goes into 250 six times (6*41=246). Subtract 246 from 250, we get 4. Bring down another zero: 40. 41 goes into 40 zero times. Bring down another zero: 400. 41 goes into 400 nine times (9*41=369). Subtract 369 from 400, we get 31. Bring down another zero: 310. 41 goes into 310 seven times (7*41=287). Subtract 287 from 310, we get 23. Now, we're back to where we started with 23.So, the decimal repeats. So, 23/41 is approximately 0.56097560975..., so approximately 0.561. So, the maximum likelihood estimate for ( p ) is about 0.561.Wait, but let me think again. Is this the correct approach? I remember that for maximum likelihood estimation in a binomial model, the MLE for ( p ) is indeed ( hat{p} = frac{k}{n} ). So, yes, this should be correct.So, that's the first part done. Now, moving on to the second part.The Blues scored an average of 3 goals per game, and the number of goals follows a Poisson distribution. I need to find the probability that in a randomly chosen game, they scored exactly 5 goals, and also find the variance of the number of goals scored in a single game.Okay, so Poisson distribution is used to model the number of events happening in a fixed interval of time or space, and it's characterized by a single parameter ( lambda ), which is the average rate (mean) of occurrence. In this case, the average number of goals per game is 3, so ( lambda = 3 ).First, let's recall the formula for the Poisson probability mass function:[P(X = k) = frac{lambda^k e^{-lambda}}{k!}]Where ( X ) is the number of occurrences (goals, in this case), ( k ) is the number of occurrences we're interested in, ( lambda ) is the average rate, and ( e ) is the base of the natural logarithm.So, to find the probability that the Blues scored exactly 5 goals in a game, we plug in ( k = 5 ) and ( lambda = 3 ):[P(X = 5) = frac{3^5 e^{-3}}{5!}]Let me compute that step by step.First, compute ( 3^5 ). 3^1 is 3, 3^2 is 9, 3^3 is 27, 3^4 is 81, 3^5 is 243.Next, compute ( e^{-3} ). I know that ( e ) is approximately 2.71828, so ( e^{-3} ) is approximately 1/(2.71828)^3. Let me compute that.First, compute ( e^3 ). 2.71828^3. Let me compute 2.71828 * 2.71828 first. 2.71828 * 2.71828 is approximately 7.38906. Then, multiply that by 2.71828 again: 7.38906 * 2.71828. Let me approximate this.7 * 2.71828 is about 19.02796, 0.38906 * 2.71828 is approximately 1.056. So, adding together, approximately 19.02796 + 1.056 ≈ 20.084. So, ( e^3 ) is approximately 20.0855. Therefore, ( e^{-3} ) is approximately 1/20.0855 ≈ 0.049787.So, ( e^{-3} ) ≈ 0.049787.Next, compute 5! (5 factorial). 5! = 5 * 4 * 3 * 2 * 1 = 120.So, putting it all together:[P(X = 5) = frac{243 * 0.049787}{120}]First, compute 243 * 0.049787. Let me do that.243 * 0.049787. Let's break it down:243 * 0.04 = 9.72243 * 0.009787 ≈ 243 * 0.01 = 2.43, so subtract 243 * (0.01 - 0.009787) = 243 * 0.000213 ≈ 0.0518. So, 2.43 - 0.0518 ≈ 2.3782.So, total is approximately 9.72 + 2.3782 ≈ 12.0982.Therefore, 243 * 0.049787 ≈ 12.0982.Now, divide that by 120:12.0982 / 120 ≈ 0.100818.So, approximately 0.1008, or 10.08%.Wait, let me check that calculation again because 243 * 0.049787 is approximately 12.0982, and 12.0982 divided by 120 is approximately 0.1008. So, yes, that seems correct.Alternatively, maybe I can compute it more accurately.Compute 243 * 0.049787:First, 243 * 0.04 = 9.72243 * 0.009787:Compute 243 * 0.009 = 2.187243 * 0.000787 ≈ 243 * 0.0007 = 0.1701, and 243 * 0.000087 ≈ 0.021141So, total is approximately 2.187 + 0.1701 + 0.021141 ≈ 2.378241So, total is 9.72 + 2.378241 ≈ 12.098241Divide by 120: 12.098241 / 120 = 0.100818675So, approximately 0.1008, or 10.08%.So, the probability is approximately 10.08%.Wait, but let me think if I did that correctly. Alternatively, maybe I can use more precise values.Alternatively, perhaps I can use a calculator for more precision, but since I'm doing this manually, let's see.Alternatively, maybe I can use the exact formula:( P(X = 5) = frac{3^5 e^{-3}}{5!} )We can compute 3^5 = 243, 5! = 120, so 243 / 120 = 2.025.So, 2.025 * e^{-3} ≈ 2.025 * 0.049787 ≈ 0.1008.Yes, same result.So, the probability is approximately 0.1008, or 10.08%.Now, the second part of the second question is to find the variance of the number of goals scored in a single game.I remember that for a Poisson distribution, the variance is equal to the mean. So, if the average number of goals is 3, then the variance is also 3.Wait, is that correct? Let me recall: in a Poisson distribution, the variance is equal to the mean, which is ( lambda ). So, yes, if ( lambda = 3 ), then the variance is 3.So, the variance is 3.Wait, but let me think again. Is that true? Because sometimes people confuse variance and standard deviation, but in Poisson, variance is equal to the mean. So, yes, that's correct.So, to recap:1. The maximum likelihood estimate of ( p ) is 46/82, which simplifies to 23/41, approximately 0.561.2. The probability of scoring exactly 5 goals is approximately 0.1008, and the variance is 3.Wait, but let me just make sure I didn't make any calculation errors, especially in the Poisson probability.Alternatively, maybe I can use more precise values.Compute ( e^{-3} ) more accurately. Let's use more decimal places.( e ) is approximately 2.718281828459045.So, ( e^3 ) is approximately 20.0855369232.Therefore, ( e^{-3} ) is approximately 1 / 20.0855369232 ≈ 0.0497870683678.So, ( e^{-3} ≈ 0.0497870683678 ).So, 3^5 is 243, as before.5! is 120.So, ( P(X = 5) = (243 * 0.0497870683678) / 120 ).Compute 243 * 0.0497870683678:Let me compute 243 * 0.0497870683678.First, 243 * 0.04 = 9.72243 * 0.0097870683678:Compute 243 * 0.009 = 2.187243 * 0.0007870683678 ≈ 243 * 0.0007 = 0.1701, and 243 * 0.0000870683678 ≈ 0.02116.So, total is approximately 2.187 + 0.1701 + 0.02116 ≈ 2.37826.So, total is 9.72 + 2.37826 ≈ 12.09826.Divide by 120: 12.09826 / 120 ≈ 0.1008188333.So, approximately 0.1008188333, which is about 0.1008 or 10.08%.So, that's consistent with my earlier calculation.Therefore, I think that's correct.So, summarizing:1. The maximum likelihood estimate of ( p ) is 46/82, which simplifies to 23/41, approximately 0.561.2. The probability of scoring exactly 5 goals is approximately 0.1008, and the variance is 3.Wait, but let me just make sure I didn't make any mistakes in simplifying 46/82.46 divided by 82: both are even numbers, so divide numerator and denominator by 2: 23/41. 23 is a prime number, so it can't be reduced further. So, 23/41 is the simplified fraction, which is approximately 0.560975609756..., so 0.561 when rounded to three decimal places.Yes, that seems correct.So, I think I've got both parts figured out. The maximum likelihood estimate is 23/41, and the probability of scoring exactly 5 goals is approximately 0.1008, with a variance of 3.Final Answer1. The maximum likelihood estimate of ( p ) is boxed{dfrac{23}{41}}.2. The probability of scoring exactly 5 goals is approximately boxed{0.1008}, and the variance is boxed{3}.</think>