Final answer:
The probability that exactly 12 out of 20 children sampled go out trick-or-treating can be calculated using the binomial probability formula. This involves finding the number of ways to choose 12 children from 20 and then using the probability of success (55%) to find the exact probability.
Step-by-step explanation:
To find the probability that exactly 12 out of 20 children sampled go out trick-or-treating, when it is known that 55% of kids go trick-or-treating, we use the binomial probability formula. The formula is as follows:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of having exactly k successes in n trials.
- C(n, k) is the combination of n items taken k at a time.
- p is the probability of success on a given trial.
- n is the number of trials.
- k is the number of successes.
In this case, n = 20, k = 12, and p = 0.55. The combinations of 20 items taken 12 at a time is calculated using a calculator or combinations formula, and the rest of the terms are raised to their respective powers. Multiplying these together will give the probability that exactly 12 out of 20 children go trick-or-treating.