Statistics help

Question

Job Elimination: In the past year,13% of businesses have eliminated jobs. If 4 businesses are selected at random, find the probability that at least 2 have eliminated jobs during the last year. Round your answer to at least 3 decimal places. Do not round your immediate calculations.

Answer 1

The question is a bit ambiguous; it sounds like selection with replacement, but you aren't given the total population size (number of businesses). Since you know only the percent of businesses that eliminated jobs, you'll have to compute this as selection with replacement (independent selection).

If you know about Binomial Distributions and have access to calculator that will compute them, you're looking for P(X >=2 ) in the distribution B(4, 0.13). If not...

Suppose that when you choose a single object (business) from your population, the probability that object has a particular quality is equal to p (such an object is called a "success", even though in this case a "success" would be that the business eliminated jobs; terminology can be non-intuitive). When you make n independent selections (n=4 in this problem), the probability that exactly k of the objects selected are "successes" is given by the formula:

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

where C(n,k) = n! / (k! * (n-k)!). For this question, you'd need at least 2 "successes", so you'll have to compute P(2) + P(3) + P(4). For example:

P(2) = C(4,2) * (.13)^2 * (.87)^2 = 4!/(2!*2!) * (.0169) * (.7569) = 6 * (.0169) * (.7569) = 0.07674966

The other values are done similarly; I get P(3) = .00765... and P(4) = .00029... using technology.