This problem can be solved using the Negative Binomial Distribution. The Negative Binomial Distribution describes the probability of having to wait a certain number of trials until achieving a fixed number of successes.

a. To calculate the probability that she needs to go through 85 students to obtain the sample of 6, we use the formula for the Negative Binomial Distribution:

P(X = k) = C(k-1, r-1) * (p^r) * ((1-p)^(k-r))

where:
- X is the number of trials needed to achieve r successes,
- k is the total number of trials,
- r is the number of successes,
- p is the probability of success on each trial,
- C(k-1, r-1) is the combination of (k-1) things taken (r-1) at a time.

In this case, k = 85 (total number of students she goes through), r = 6 (number of students who obtained between 50% and 70%), and p = 0.13 (probability that a student obtained between 50% and 70%).

So, P(X = 85) = C(84, 5) * (0.13^6) * ((1-0.13)^(85-6))

b. To calculate the probability that she needs to go through at least 50 students to obtain the sample of 6, we need to find 1 minus the cumulative probability of needing to go through 49 or fewer students. This can be calculated by summing the probabilities of needing to go through exactly r students for r from 6 to 49, and subtracting this sum from 1.

So, P(X = 50) = 1 - Σ[P(X = r) for r = 6 to 49]

where P(X = r) can be calculated using the same formula as in part a.

Please note that these calculations require a calculator or software capable of computing combinations and powers.

Question

This problem can be solved using the Negative Binomial Distribution. The Negative Binomial Distribution describes the probability of having to wait a certain number of trials until achieving a fixed number of successes.

a. To calculate the probability that she needs to go through 85 students to obtain the sample of 6, we use the formula for the Negative Binomial Distribution:

P(X = k) = C(k-1, r-1) * (p^r) * ((1-p)^(k-r))

where:
- X is the number of trials needed to achieve r successes,
- k is the total number of trials,
- r is the number of successes,
- p is the probability of success on each trial,
- C(k-1, r-1) is the combination of (k-1) things taken (r-1) at a time.

In this case, k = 85 (total number of students she goes through), r = 6 (number of students who obtained between 50% and 70%), and p = 0.13 (probability that a student obtained between 50% and 70%).

So, P(X = 85) = C(84, 5) * (0.13^6) * ((1-0.13)^(85-6))

b. To calculate the probability that she needs to go through at least 50 students to obtain the sample of 6, we need to find 1 minus the cumulative probability of needing to go through 49 or fewer students. This can be calculated by summing the probabilities of needing to go through exactly r students for r from 6 to 49, and subtracting this sum from 1.

So, P(X >= 50) = 1 - Σ[P(X = r) for r = 6 to 49]

where P(X = r) can be calculated using the same formula as in part a.

Please note that these calculations require a calculator or software capable of computing combinations and powers.

Knowee AI · Accepted Answer