Final answer:
To calculate the probability of at least two seniors entering military service out of a randomly selected group of 20, we subtract the cumulative probability of 0 or 1 senior entering service from 1, using the binomial probability formula with a success rate of 15%.
Step-by-step explanation:
The problem is asking us to calculate the probability of at least two seniors entering military service from a randomly selected group of 20 seniors, given a known rate of 15% for the general senior population. We can tackle this problem using the binomial probability formula, since there are only two outcomes for each senior - they either enter military service or they do not - and each senior's decision is independent of the others'.
The formula for the binomial probability of exactly k successes in n trials is P(X = k) = (nCk) * (pk) * ((1-p)(n-k)), where nCk represents the number of combinations of n items taken k at a time, p is the probability of success on a single trial, and (1-p) is the probability of failure on a single trial.
However, rather than calculating the probability of exactly two seniors entering military service, we want the probability of at least two. This means we have to calculate the probability of 2 or more successes. Instead of calculating this directly, it is easier to calculate the probability of the compliment event - namely, 0 or 1 senior entering service - and subtract this from 1 to get our answer. This means we need to calculate P(X = 0) and P(X = 1) and then subtract the sum from 1.
So, the probability of finding at least two students entering military service is calculated as 1 - (P(X = 0) + P(X = 1)). We can use the above formula to find P(X = 0) and P(X = 1), plug in p = 0.15, n = 20, and sum the probabilities for k = 0 and k = 1.