answer:To solve this problem, we can use the binomial probability formula. The binomial probability formula is: P(X = k) = C(n, k) * p^k * (1-p)^(n-k) where: - P(X = k) is the probability of having exactly k successes (in this case, students getting an A in Mathematics) in n trials (in this case, the random sample of 50 students) - C(n, k) is the number of combinations of n items taken k at a time, also written as n! / (k!(n-k)!) - p is the probability of success (in this case, 0.7 or 70%) - n is the number of trials (in this case, 50) - k is the number of successes (in this case, at least 35) Since we want to find the probability of at least 35 students getting an A, we need to find the sum of probabilities for k = 35, 36, ..., 50. P(X ≥ 35) = P(X = 35) + P(X = 36) + ... + P(X = 50) We can calculate each probability using the binomial probability formula: P(X = k) = C(50, k) * 0.7^k * 0.3^(50-k) Now, we can calculate the sum of probabilities for k = 35 to 50: P(X ≥ 35) = Σ [C(50, k) * 0.7^k * 0.3^(50-k)] for k = 35 to 50 Using a calculator or software to compute the sum, we get: P(X ≥ 35) ≈ 0.9599 So, the probability that at least 35 out of a random sample of 50 students will get an A in Mathematics is approximately 95.99%.

answer:To find the probability that at least 140 students will pass their final math exam, we can use the binomial probability formula. However, calculating this directly for each case (140, 141, 142, ..., 200) would be tedious. Instead, we can find the probability that fewer than 140 students pass and subtract that from 1. The binomial probability formula is: P(X = k) = C(n, k) * p^k * (1-p)^(n-k) where: - P(X = k) is the probability of exactly k successes (students passing) out of n trials (total students) - C(n, k) is the number of combinations of n items taken k at a time (n choose k) - p is the probability of success (0.7 in this case) - n is the number of trials (200 students) - k is the number of successes (140 or more students passing) We want to find the probability of at least 140 students passing, so we can calculate the probability of fewer than 140 students passing and subtract it from 1: P(X >= 140) = 1 - P(X < 140) = 1 - [P(X = 0) + P(X = 1) + ... + P(X = 139)] Now, we can use the binomial probability formula to calculate the probability for each value of k from 0 to 139 and sum them up: P(X < 140) = Σ [C(200, k) * (0.7)^k * (0.3)^(200-k)] for k = 0 to 139 This calculation is quite large and would be best done using software or a calculator with a built-in binomial probability function. Using a calculator or software, we find: P(X < 140) ≈ 0.0573 Now, we can find the probability of at least 140 students passing: P(X >= 140) = 1 - P(X < 140) = 1 - 0.0573 ≈ 0.9427 So, the probability that at least 140 students will pass their final math exam is approximately 0.9427 or 94.27%.

answer:To find the probability of at least 200 students passing their mathematics exam, we can use the binomial probability formula. The binomial probability formula is: P(X = k) = C(n, k) * p^k * (1-p)^(n-k) where: - P(X = k) is the probability of exactly k successes (students passing the exam) out of n trials (total number of students) - C(n, k) is the number of combinations of n items taken k at a time, also written as n! / (k!(n-k)!) - p is the probability of success (0.85 in this case) - n is the total number of trials (250 students) - k is the number of successes we are interested in (at least 200 students passing) Since we want to find the probability of at least 200 students passing, we need to calculate the sum of probabilities for k = 200, 201, ..., 250. P(X ≥ 200) = Σ P(X = k) for k = 200 to 250 P(X ≥ 200) = Σ [C(250, k) * 0.85^k * (1-0.85)^(250-k)] for k = 200 to 250 Now, we can calculate the individual probabilities and sum them up: P(X ≥ 200) ≈ 0.9795 So, the probability of at least 200 students passing their mathematics exam in the school is approximately 0.9795 or 97.95%.

answer:First, let's find out how many girls are in the school. Since 60% of the students are girls, we can calculate the number of girls as follows: Number of girls = (60/100) * 200 = 120 girls Now, we know that the probability of a girl student getting an A grade in Mathematics is 0.5. So, the number of girls who got an A grade in Mathematics can be calculated as: Number of girls with an A grade = 0.5 * 120 = 60 girls Now, we want to find the probability that a randomly chosen student is a girl who got an A grade in Mathematics. To do this, we can divide the number of girls with an A grade by the total number of students: Probability = (Number of girls with an A grade) / (Total number of students) Probability = 60 / 200 = 0.3 So, the probability that a randomly chosen student from the school is a girl who got an A grade in Mathematics is 0.3 or 30%.