🧠:<think>Okay, so I have two problems here related to an Australian rules football player. Let me try to tackle them one by one.Starting with the first problem:1. The player played a total of ( n ) games, with an average score per game of ( S ). The total score over his career was 960, and he played 120 games. So, I can write that as:Total score = Average score per game × Number of games 960 = S × 120 So, solving for S, we get S = 960 / 120 = 8. So, his average score per game was 8.Wait, but the problem says that in his first season, he scored an average of 8 goals per game over 15 games. So, that seems consistent because his overall average is also 8. Hmm, maybe I need to think more about the Poisson distribution part.It says that the number of points he scored in each game follows a Poisson distribution with parameter ( lambda ), which is the average number of goals per game he scored during his first season. Since his first season average was 8, does that mean ( lambda = 8 )?But wait, the Poisson distribution is used to model the number of times an event occurs in an interval. In this case, the number of goals per game. The parameter ( lambda ) is the average rate (expected value). So, if his average per game in the first season was 8, then ( lambda ) should indeed be 8.But let me double-check. The total score over his career was 960 in 120 games, so average per game is 8. His first season average was also 8 over 15 games. So, the parameter ( lambda ) for the Poisson distribution is 8.So, I think the value of ( lambda ) is 8.Moving on to the second problem:2. After retirement, he wants to improve the average score per game by 15% compared to his playing average. His playing average was 8 goals per game, so a 15% improvement would be:Improved average = 8 + (15% of 8) 15% of 8 is 1.2, so 8 + 1.2 = 9.2.So, the improved average score per game is 9.2.He wants his team to achieve this average over 20 games. So, the total score needed would be:Total score = Improved average × Number of games Total score = 9.2 × 20 = 184.Wait, 9.2 multiplied by 20 is 184? Let me calculate that again. 9 × 20 is 180, and 0.2 × 20 is 4, so 180 + 4 = 184. Yes, that seems correct.So, the total score his team needs to achieve is 184.But let me think again if I interpreted the 15% improvement correctly. It says "improve the average score per game by 15% compared to his playing average." So, his playing average was 8, so 15% more would indeed be 8 × 1.15 = 9.2. So, that seems right.Therefore, the total score needed is 184.Final Answer1. The value of ( lambda ) is boxed{8}.2. The total score his team needs to achieve is boxed{184}.</think>

🧠:<think>Okay, so I've got these two problems to solve. The first one is about RSA encryption, and the second one is about minimizing latency using some quadratic function. Let me tackle them one by one.Starting with the first problem: I need to encrypt the transaction amount M = 25000 using RSA with the given public key (e, n) = (65537, 119308617327808310399282994092317332743). Hmm, okay, RSA encryption is done using the formula C = M^e mod n. So, I need to compute 25000 raised to the power of 65537 modulo that huge number n. That seems computationally intensive. I wonder if there's a smarter way to do this without actually calculating such a massive exponentiation manually.Wait, maybe I can use the property of modular exponentiation. I remember that exponentiation can be broken down using the method of exponentiation by squaring, which is more efficient. But even so, doing this by hand would take forever. Maybe I can use some properties or see if 25000 and n are coprime? Let me check if 25000 and n share any common factors.First, factorizing 25000: 25000 = 25 * 1000 = 5^3 * 2^3 * 5^3 = 2^3 * 5^6. So, 25000 is 2^3 * 5^6. Now, n is 119308617327808310399282994092317332743. I don't know the factors of n, but since it's an RSA modulus, it's the product of two large primes, p and q. So, unless 2 or 5 divides n, 25000 and n are coprime. But since n is a large prime product, it's unlikely that 2 or 5 divides it. So, I can proceed with the encryption.But still, calculating 25000^65537 mod n is not feasible by hand. Maybe I can use the Chinese Remainder Theorem if I know p and q, but I don't have them. Alternatively, perhaps I can use Euler's theorem, but that also requires knowing φ(n), which I don't have. So, maybe I need to accept that I can't compute this manually and perhaps just state the formula or use a computational tool. But since this is a theoretical problem, maybe I can just write the expression.Wait, maybe the problem expects me to recognize that calculating such a large exponent is impractical without a computer, so perhaps the answer is just the expression C = 25000^65537 mod 119308617327808310399282994092317332743. But I'm not sure. Maybe I can compute it modulo n by breaking it down, but I don't see an easy way.Alternatively, perhaps the modulus n is a prime? Let me check if n is a prime number. Wait, n is 119308617327808310399282994092317332743. That's a 39-digit number. I don't know if it's prime. If it's prime, then φ(n) = n - 1, but again, without knowing, I can't proceed. So, maybe the answer is just the expression as is.Moving on to the second problem: minimizing the processing time T(x) = 5x^2 - 20x + 60. This is a quadratic function in terms of x, and since the coefficient of x^2 is positive, it opens upwards, meaning the vertex is the minimum point. The x-coordinate of the vertex is at -b/(2a). Here, a = 5 and b = -20. So, x = -(-20)/(2*5) = 20/10 = 2. So, the optimal number of transactions per second is 2.Then, plugging x = 2 back into T(x): T(2) = 5*(2)^2 - 20*(2) + 60 = 5*4 - 40 + 60 = 20 - 40 + 60 = 40. So, the minimum processing time is 40 milliseconds.Wait, but let me double-check that. The function is T(x) = 5x² -20x +60. The derivative is T’(x) = 10x -20. Setting derivative to zero: 10x -20 =0 => x=2. So, yes, x=2 is correct. Then T(2)=5*4 -40 +60=20-40+60=40. Yep, that seems right.So, for the first problem, I think I have to leave it as C = 25000^65537 mod n, since computing it manually isn't feasible. For the second problem, the optimal x is 2 and the minimum T is 40 ms.Final Answer1. The ciphertext ( C ) is boxed{25000^{65537} mod 119308617327808310399282994092317332743}.2. The optimal number of transactions per second is (boxed{2}) and the minimum processing time is (boxed{40}) milliseconds.</think>

🧠:<think>Alright, so I have this problem about analyzing customer feedback to identify potential defects in a software product. The support agent has collected feedback from 1000 customers, each giving a rating from 1 to 5 for two aspects: usability and reliability. The data is structured as a matrix F, where each entry F_ij is the feedback from the ith customer on the jth aspect. The first part asks me to define the defect score D for a particular aspect as the sum of squared deviations of customer ratings from the mean rating, normalized by the number of customers. Then, I need to calculate the defect scores D1 and D2 for usability and reliability, respectively, and provide a general expression in terms of the matrix F.Okay, so let me break this down. The defect score is defined as the sum of squared deviations from the mean, normalized by the number of customers. That sounds familiar—it's essentially the variance of the ratings for each aspect. Variance is calculated as the average of the squared differences from the mean. So, if I can compute the variance for each aspect, that should give me the defect scores D1 and D2.Given that F is a matrix with 1000 customers and 2 aspects, each aspect will have 1000 ratings. For each aspect j (where j=1 for usability and j=2 for reliability), the defect score Dj is the variance of the ratings in column j of matrix F.So, mathematically, for each aspect j, the mean rating μj is the average of all F_ij for that aspect. Then, the defect score Dj is the sum over all customers i of (F_ij - μj)^2, divided by the number of customers, which is 1000.Expressed in terms of the matrix F, for each column j, compute the mean μj, then compute the squared deviations from μj for each entry in column j, sum them up, and divide by 1000.So, in formula terms, for each j (1 and 2):μj = (1/1000) * Σ (from i=1 to 1000) F_ijDj = (1/1000) * Σ (from i=1 to 1000) (F_ij - μj)^2Therefore, the general expression for D1 and D2 in terms of F is as above.Wait, let me make sure. The problem says "sum of squared deviations... normalized by the number of customers." So that's exactly the variance. So, yes, D1 is the variance of the first column, D2 is the variance of the second column.So, in terms of F, it's the variance of each column.Moving on to part 2. It says, assuming that the defect scores D1 and D2 are normally distributed, with means μ1 = 1.5 and μ2 = 2.0, and standard deviations σ1 = 0.3 and σ2 = 0.4, respectively, determine the probability that both defect scores exceed 2.5.Hmm, okay. So, D1 ~ N(1.5, 0.3^2) and D2 ~ N(2.0, 0.4^2). We need to find P(D1 > 2.5 and D2 > 2.5).Assuming that D1 and D2 are independent? The problem doesn't specify, but in the absence of information, I think we can assume independence unless stated otherwise.So, if they are independent, then the joint probability is the product of the individual probabilities.So, first, find P(D1 > 2.5) and P(D2 > 2.5), then multiply them together.To compute these probabilities, we can standardize the normal variables.For D1: Z1 = (D1 - μ1)/σ1 = (2.5 - 1.5)/0.3 = 1.0 / 0.3 ≈ 3.3333Similarly, for D2: Z2 = (D2 - μ2)/σ2 = (2.5 - 2.0)/0.4 = 0.5 / 0.4 = 1.25Then, P(D1 > 2.5) = P(Z1 > 3.3333) and P(D2 > 2.5) = P(Z2 > 1.25)Looking up these Z-scores in the standard normal distribution table.For Z1 = 3.3333, which is approximately 3.33. The standard normal table gives the probability that Z is less than a certain value. So, P(Z > 3.33) is 1 - P(Z < 3.33). From tables, P(Z < 3.33) is about 0.9995, so P(Z > 3.33) ≈ 0.0005.For Z2 = 1.25, P(Z < 1.25) is about 0.8944, so P(Z > 1.25) ≈ 1 - 0.8944 = 0.1056.Therefore, P(D1 > 2.5 and D2 > 2.5) ≈ 0.0005 * 0.1056 ≈ 0.0000528.So, approximately 0.00528%, which is a very small probability.Wait, let me double-check the Z-scores.For D1: (2.5 - 1.5)/0.3 = 1.0 / 0.3 ≈ 3.3333. Yes, that's correct.For D2: (2.5 - 2.0)/0.4 = 0.5 / 0.4 = 1.25. Correct.Looking up Z=3.3333, the cumulative probability is indeed very close to 1, so the tail probability is about 0.0005.For Z=1.25, the cumulative is about 0.8944, so the tail is 0.1056.Multiplying them gives 0.0005 * 0.1056 ≈ 0.0000528.So, the probability is approximately 0.00528%, which is 5.28e-5.Alternatively, using more precise calculations:For Z=3.3333, the exact probability can be found using the standard normal CDF. Using a calculator or precise table, P(Z > 3.33) is approximately 0.000427.Similarly, for Z=1.25, P(Z > 1.25) is approximately 0.1056.Multiplying these gives 0.000427 * 0.1056 ≈ 0.00004505, which is about 0.0045%.So, roughly 0.0045% chance that both defect scores exceed 2.5.Therefore, the probability is approximately 0.000045, or 4.5e-5.I think that's the answer.Final Answer1. The defect scores are given by ( D_j = frac{1}{1000} sum_{i=1}^{1000} (F_{ij} - mu_j)^2 ) for ( j = 1, 2 ). Thus, ( D_1 = boxed{frac{1}{1000} sum_{i=1}^{1000} (F_{i1} - mu_1)^2} ) and ( D_2 = boxed{frac{1}{1000} sum_{i=1}^{1000} (F_{i2} - mu_2)^2} ).2. The probability that both defect scores exceed 2.5 is approximately ( boxed{0.000045} ).</think>

🧠:<think>Alright, so I've got these two problems to solve related to designing a secure facility. Let me take them one at a time. Starting with the first one: Biometric Access Control Optimization. The system uses both fingerprint and retinal scans. The probabilities given are 0.999 for the fingerprint scan being accurate and 0.998 for the retinal scan. The system allows access only if both scans are accurate. I need to find the probability that an authorized person is granted access on the first attempt.Hmm, okay. So since both scans need to be accurate, I think I need to multiply their individual probabilities. Because for independent events, the probability of both happening is the product of their probabilities. So, that would be 0.999 multiplied by 0.998. Let me calculate that.0.999 * 0.998. Let me do this step by step. 0.999 * 0.998 is the same as (1 - 0.001) * (1 - 0.002). Expanding this, it's 1 - 0.001 - 0.002 + 0.000002. So, 1 - 0.003 + 0.000002. That's 0.997 + 0.000002, which is 0.997002. Wait, let me verify that multiplication another way. 0.999 * 0.998. Let's compute 999 * 998 first. 999 * 998 is (1000 - 1)(1000 - 2) = 1000*1000 - 1000*2 - 1000*1 + 1*2 = 1,000,000 - 2,000 - 1,000 + 2 = 1,000,000 - 3,000 + 2 = 997,002. So, 0.999 * 0.998 is 0.997002. Yep, that matches. So, the probability is 0.997002, which is approximately 0.997 or 99.7%.Okay, that seems straightforward. I think that's the answer for the first part.Moving on to the second problem: Encryption Key Distribution. The encryption key is distributed among 5 security officers, each holding a unique part. At least 3 parts are needed to reconstruct the key. The probability of any single officer being unavailable is 0.2. I need to find the probability that the full key can be reconstructed at any given time.So, this sounds like a problem involving combinations and probabilities. Since at least 3 officers need to be available, we can think of it as the probability that 3, 4, or all 5 officers are available. Alternatively, it's 1 minus the probability that fewer than 3 are available (i.e., 0, 1, or 2 officers are available). But let me think about it more carefully. Each officer has a 0.2 chance of being unavailable, so the probability of being available is 1 - 0.2 = 0.8.We can model this using the binomial probability formula. The probability of exactly k officers being available out of n is C(n, k) * p^k * (1-p)^(n-k). Here, n=5, p=0.8, and we need the sum for k=3,4,5.Alternatively, it's easier to compute 1 minus the sum for k=0,1,2. Let me compute both ways to verify.First, let's compute the probability that at least 3 are available:P(at least 3) = P(3) + P(4) + P(5)Where P(k) = C(5, k) * (0.8)^k * (0.2)^(5 - k)Let me compute each term:P(3) = C(5,3) * (0.8)^3 * (0.2)^2C(5,3) is 10.So, 10 * (0.512) * (0.04) = 10 * 0.02048 = 0.2048P(4) = C(5,4) * (0.8)^4 * (0.2)^1C(5,4) is 5.So, 5 * (0.4096) * (0.2) = 5 * 0.08192 = 0.4096P(5) = C(5,5) * (0.8)^5 * (0.2)^0C(5,5) is 1.So, 1 * (0.32768) * 1 = 0.32768Adding these up: 0.2048 + 0.4096 + 0.327680.2048 + 0.4096 = 0.61440.6144 + 0.32768 = 0.94208Alternatively, computing 1 - [P(0) + P(1) + P(2)]Let's compute P(0):P(0) = C(5,0) * (0.8)^0 * (0.2)^5 = 1 * 1 * 0.00032 = 0.00032P(1) = C(5,1) * (0.8)^1 * (0.2)^4 = 5 * 0.8 * 0.0016 = 5 * 0.00128 = 0.0064P(2) = C(5,2) * (0.8)^2 * (0.2)^3 = 10 * 0.64 * 0.008 = 10 * 0.00512 = 0.0512Adding these: 0.00032 + 0.0064 + 0.0512 = 0.05792So, 1 - 0.05792 = 0.94208Same result. So, the probability is 0.94208, which is approximately 0.9421 or 94.21%.Wait, let me double-check my calculations to make sure I didn't make any arithmetic errors.Starting with P(3):C(5,3)=10, (0.8)^3=0.512, (0.2)^2=0.04. 10*0.512=5.12, 5.12*0.04=0.2048. Correct.P(4):C(5,4)=5, (0.8)^4=0.4096, (0.2)^1=0.2. 5*0.4096=2.048, 2.048*0.2=0.4096. Correct.P(5):C(5,5)=1, (0.8)^5=0.32768, (0.2)^0=1. So, 1*0.32768=0.32768. Correct.Sum: 0.2048 + 0.4096 = 0.6144; 0.6144 + 0.32768 = 0.94208. Correct.Alternatively, computing the complement:P(0)=1*1*(0.2)^5=0.00032P(1)=5*(0.8)*(0.2)^4=5*0.8*0.0016=5*0.00128=0.0064P(2)=10*(0.8)^2*(0.2)^3=10*0.64*0.008=10*0.00512=0.0512Sum: 0.00032 + 0.0064=0.00672; 0.00672 + 0.0512=0.057921 - 0.05792=0.94208. Correct.So, both methods give the same result, so I think that's solid.Therefore, the probability that the full encryption key can be reconstructed is approximately 0.9421 or 94.21%.I think that's it. Both problems seem to be solved correctly.Final Answer1. The probability of being granted access is boxed{0.9970}.2. The probability that the full encryption key can be reconstructed is boxed{0.9421}.</think>