I'm doing practice problems for a test on probability and I'm stuck on this one.

An archer hits a target 1/3 of the time, using binomial theorem, tell the probability that he hits the target 10 times on 50 shots.

Can you please tell me how to get the answer and explain thoroughly please? Sorry, I have a really bad teacher and we never went over how to use binomial theorem with probability.

If he takes 50 shots, we can write the binomial as

\(\displaystyle (p+q)^n=\left(p+q\right)^{50}\) where p=probability of a hit, q=probability of a miss.

\(\displaystyle =\binom{50}{0}p^0q^{50}+\binom{50}{1}p^1q^{49}+\binom{50}{2}p^2q^{48}+\binom{50}{3}p^3q^{47}+.......\)

sums the probabilities of no hits (which is 50 misses), that's the first term in the expansion,

one hit (49 misses), that's the 2nd term,

two hits (48 misses), this is the third term etc etc

The \(\displaystyle 50c_k\) values are the number of ways that number of "k" hits can happen.

Hence 10 hits includes 40 misses and there are \(\displaystyle \binom{50}{10}=\binom{50}{40}\) ways this can happen.

Hence the probability of exactly 10 hits is

\(\displaystyle \binom{50}{10}\left(\frac{1}{3}\right)^{10}\left(\frac{2}{3}\right)^{40}\)