Final answer:
To find the probability of four or five workers out of ten being union members, one must use the binomial probability formula, calculate individually for four and five union members, and then sum the probabilities.
Step-by-step explanation:
To calculate the probability that either four or five workers out of ten selected randomly are members of a labor union, given that two in five are union members and three in five are not, we use the binomial probability formula:
P(X = k) = C(n, k) × (p^k) × ((1-p)^(n-k))
Where:
- C(n, k) is the combination of n items taken k at a time.
- p is the probability of success (being a union member).
- n is the total number of trials (employees selected).
- k is the total number of successes (union members selected).
In this case, p is 2/5, n is 10, and we need to find the probabilities for k being 4 and k being 5.
First, calculate the probabilities individually and then sum them to find the overall probability.
The probability of exactly four union members being selected:
P(X = 4) = C(10, 4) × (2/5)^4 × (3/5)^6
The probability of exactly five union members being selected:
P(X = 5) = C(10, 5) × (2/5)^5 × (3/5)^5
Sum P(X = 4) and P(X = 5) to obtain the final probability.
The calculations will yield the result, giving the probability that four or five workers out of the ten randomly selected are members of the union.