Final answer:
The probability that exactly 2 out of 3 randomly picked hotel guests are NOT there for personal reasons, given a 30% chance that a guest is there for personal reasons, is calculated using a binomial probability formula. The result is approximately 0.441, which is not among the provided options, suggesting a possible error in the options given.
Step-by-step explanation:
The student question pertains to calculating the probability that exactly 2 out of 3 randomly picked hotel guests are NOT there for personal reasons, given that 30% are there for personal reasons. This is a binomial probability problem since there are only two possible outcomes for each trial (guest is there for personal reasons or not), a fixed number of trials (3 guests), and the probability of success (a guest being there for personal reasons) remains constant (30% or 0.30).
The probability that a guest is NOT there for personal reasons is 1 - 0.30 = 0.70 or 70%. We want to find the probability that exactly 2 guests out of 3 are NOT there for personal reasons. Using the binomial probability formula:
P(x) = (nCx)(p^x)(1-p)^(n-x), where:
- n = number of trials (3 guests)
- x = number of successes (in this case, 'success' means a guest is NOT there for personal reasons, so x=2)
- p = probability of success on any given trial (70%)
We calculate: P(2) = 3C2 * (0.70^2) * (0.30^1) = 3 * 0.49 * 0.30 = 0.441. Therefore, the answer is approximately 0.441, which is not an option provided. The correct calculation, however, seems to indicate an error in the question options given.