Question

In a large shipping company, 70% of packages arrive at their destination on time. If 9 packages are selected randomly, what is the probability that exactly 6 of them arrive at their destination on time?

a. 26.68%
b. 46.28%
c. 53.72%
d. 66.67%

Answer 1

To find the probability that exactly 6 out of 9 packages arrive on time, we can use the binomial probability formula. The correct answer is option a. 26.68%

Step-by-step explanation:

To find the probability that exactly 6 out of 9 packages arrive on time, we can use the binomial probability formula:

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

where:

• P(X=k) is the probability of getting exactly k successes
• C(n, k) is the number of ways to choose k successes out of n trials
• p is the probability of success
• n is the number of trials

In this case, n=9, k=6, and p=0.7. Plugging these values into the formula:

P(X=6) = C(9, 6) * 0.7^6 * (1-0.7)^(9-6)

Using the combination formula C(n, k) = n! / (k!(n-k)!), we can calculate:

P(X=6) = (9! / (6!(9-6)!)) * 0.7^6 * 0.3^3

P(X=6) = (9! / (6!3!)) * 0.7^6 * 0.3^3

P(X=6) = (9 * 8 * 7 / (3 * 2 * 1)) * 0.7^6 * 0.3^3

P(X=6) = 84 * 0.7^6 * 0.3^3

After evaluating this expression, the probability comes out to be approximately 26.68%. Therefore, the correct answer is (a) 26.68%.