An archer hits a target with a probability of 0.65. Assuming independence, what would be the probability of getting exactly 8 hits out of 12 trials.

This is a binomial probability problem. The binomial probability formula is:

P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))

where:

• P(X=k) is the probability of 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
• n is the number of trials
• k is the number of successes

In this case, p=0.65 (probability of hitting the target), n=12 (number of trials), and k=8 (number of hits).

First, calculate C(n, k), the number of combinations of 12 items taken 8 at a time. This can be calculated as:

C(n, k) = n! / [k!(n-k)!]

where "!" denotes factorial, which is the product of all positive integers up to that number. So,

C(12, 8) = 12! / [8!(12-8)!] = 495

Next, calculate p^k, the probability of success to the power of the number of successes:

p^k = 0.65^8 = 0.0173

Then, calculate (1-p)^(n-k), the probability of failure to the power of the number of failures:

(1-p)^(n-k) = (1-0.65)^(12-8) = 0.35^4 = 0.0150

Finally, multiply these three values together to get the probability of exactly 8 hits in 12 trials:

P(X=8) = C(n, k) * (p^k) * ((1-p)^(n-k)) = 495 * 0.0173 * 0.0150 = 0.122

So, the probability of getting exactly 8 hits out of 12 trials is approximately 0.122 or 12.2%.