Suppose that you are in a class of 31 students and it is assumed that approximately 13% of the population is left-handed. (Give your answers correct to three decimal places.)

(a) Compute the probability that exactly five students are left-handed.


(b) Compute the probability that at most four students are left-handed.


(c) Compute the probability that at least six students are left-handed.

35

0.45

To compute the probabilities in this scenario, we can use the binomial probability formula. The binomial probability formula is given by:

P(X = k) = nCk * p^k * (1-p)^(n-k)

Where:
- P(X = k) is the probability of getting exactly k successes
- n is the number of trials or observations
- p is the probability of success in a single trial
- nCk is the number of combinations of n items taken k at a time
- ^ represents exponentiation

(a) To compute the probability that exactly five students are left-handed:

In this case, n = 31 (number of students) and p = 0.13 (probability of a student being left-handed).
We want to find the probability of exactly 5 students being left-handed, so k = 5.

Using the binomial probability formula, we have:

P(X = 5) = 31C5 * (0.13)^5 * (1-0.13)^(31-5)

= (31! / (5! * (31-5)!)) * (0.13^5) * (0.87^26)

Calculating this expression will yield the probability.

(b) To compute the probability that at most four students are left-handed:

To find this probability, we need to sum up the individual probabilities of having 0, 1, 2, 3, or 4 left-handed students.

P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

Using the binomial formula with different values of k, you can calculate each term and then add them together to obtain the desired probability.

(c) To compute the probability that at least six students are left-handed:

To calculate the probability of "at least" a certain number of left-handed students, we need to find the complement of the probability of having fewer than six left-handed students.

P(X ≥ 6) = 1 - P(X ≤ 5)

We can use the same approach as in part (b) to calculate the complementary probability and subtract it from 1 to find the desired probability.

By following these steps and performing the necessary calculations, you can find the probabilities for each scenario.