Suppose that you are in a class of 31 students and it is assumed that approximately 10% 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.
To compute the probabilities, we can use the binomial distribution formula. The binomial distribution is used when there are two possible outcomes (success or failure) in a fixed number of independent trials, and each trial has the same probability of success.
In this case:
- The number of students (n) is 31.
- The probability of a student being left-handed (p) is 0.10 (10%).
(a) To compute the probability that exactly five students are left-handed, we use the binomial probability formula:
P(X = k) = (nCk) * p^k * (1 - p)^(n - k)
where nCk denotes the number of combinations of n items taken k at a time.
Plugging in the values:
P(X = 5) = (31C5) * (0.10)^5 * (1 - 0.10)^(31 - 5)
To evaluate this expression, we need to calculate (31C5), which represents the number of combinations of 31 items taken 5 at a time. This can be calculated using the formula:
(31C5) = 31! / (5! * (31 - 5)!)
where "!" denotes the factorial of a number.
Using a calculator or computer, calculate the values of (31C5), (0.10)^5, and (1 - 0.10)^(31 - 5), and then multiply them together to find the probability P(X = 5).
(b) To compute the probability that at most four students are left-handed, we need to calculate the probabilities for each possible outcome up to four students and sum them together. We can use the binomial probability formula for each outcome and add the results.
P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
Calculate each term using the binomial probability formula as shown in part (a), and then sum them together to get P(X ≤ 4).
(c) To compute the probability that at least six students are left-handed, we need to calculate the probabilities for each possible outcome from six students to the maximum possible number of left-handed students (31). We can use the binomial probability formula for each outcome and subtract the result from 1.
P(X ≥ 6) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5))
Calculate each term using the binomial probability formula as shown in part (a), sum them together, and subtract the result from 1 to get P(X ≥ 6).