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# 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.

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## Answer to a math question 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.

Nash
4.9
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

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