Of the members of a Boy Scout troop, 15% have received the first aid merit badge. If 40 boy scouts are selected at random, find the probability that four or more will have the
First aid merit badge?
A) 86.6%
B) 36.6%
C) 81.3%
D) 31.3%

Question

Quizplus · Accepted Answer

This problem can be solved using the binomial probability formula, which is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successful trials, p is the probability of success on a single trial, and (n choose k) is the binomial coefficient. However, calculating the probability of exactly 4, 5, ..., 40 scouts having the badge directly would be cumbersome. Instead, it's easier to calculate the probability of 0, 1, 2, or 3 scouts having the badge and subtracting from 1 to find the probability of 4 or more scouts having the badge.Given that 15% of the scouts have received the first aid merit badge, p = 0.15. The probability of a scout not having the badge is 1 - p = 0.85. For n = 40 scouts, we calculate the probability of having 0, 1, 2, and 3 scouts with the badge and subtract the sum from 1.The calculation for each of these probabilities directly would be tedious, but using a binomial probability calculator or statistical software simplifies this process. The cumulative probability of having 3 or fewer scouts with the badge is found, and subtracting this from 1 gives the probability of having 4 or more scouts with the badge.After performing these calculations (or using a binomial distribution calculator), the result is approximately 86.6%, which means there's an 86.6% chance that 4 or more scouts out of a randomly selected group of 40 will have the first aid merit badge.

Of the Members of a Boy Scout Troop, 15% Have