Final answer:
The probability that exactly 2 out of 5 randomly chosen workers are smokers in a factory where 30% are smokers is found using the binomial probability formula. The calculation yields a probability of 0.3087, which is answer choice (a).
Step-by-step explanation:
The question asks about the probability that there are exactly 2 smokers in a randomly chosen group of 5 workers from a factory where 30% of employees are smokers. This is a problem that can be solved using the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where:
- n is the number of trials (5 workers)
- k is the number of successful outcomes (2 smokers)
- p is the probability of success on an individual trial (0.30 for a smoker)
- 1-p is the probability of failure on an individual trial (0.70 for a non-smoker)
The binomial coefficient (n choose k) is the number of ways to choose k successes from n trials, which can be calculated as n! / (k! * (n-k)!).
For n = 5 and k = 2:
- 5 choose 2 is 10 (there are 10 ways to pick 2 smokers out of 5 workers)
- p^k is 0.30^2
- (1-p)^(n-k) is 0.70^3
Putting these values into the formula gives us:
P(X = 2) = 10 * (0.30)^2 * (0.70)^3 = 0.3087
Therefore, the probability is 0.3087, which corresponds to answer choice (a).