A recent survey found that 82​% of​ first-year college students were involved in volunteer work at least occasionally. Suppose a random sample of 12 college students is taken. Find the probability that exactly 7 students volunteered at least occasionally.

This is a binomial probability problem where the probability of success (a student volunteering) is 82% or 0.82.

The formula for calculating the probability of exactly k successes in n trials is given by:
P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k}

Where
n = 12 (total number of trials - 12 college students),
k = 7 (number of successful trials - 7 students volunteered),
p = 0.82 (probability of success - a student volunteering),
1-p = 1-0.82 = 0.18 (probability of failure - a student not volunteering).

Plugging these values into the formula:
P(X = 7) = \binom{12}{7} \times 0.82^{7} \times 0.18^{5}
P(X = 7) = \frac{12!}{7!(12-7)!} \times 0.82^{7} \times 0.18^{5}

Calculating the values:
P(X = 7) = \frac{12!}{7!5!} \times 0.82^{7} \times 0.18^{5}
P(X = 7) = 792 \times 0.00261201 \times 0.07319778


Therefore, the probability that exactly 7 out of 12 students volunteered at least occasionally is approximately 0.037306514365531