Question

The probability of winning a match for team A is 0.6. Find the probability of winning 3 matches out of 5.

Answer 1

We have a Bernoulli trial - exactly two possible outcomes, "success" (team A wins the match) and "failure" (team A doesn't win the match) and the probability of success is the same every time the experiment is conducted (team A plays a match), $p=0.6, q=1-0.6=0.4, n=5.$

The probability that team A wins k matches out of 5 is

$P(X=k)=\begin{pmatrix}n\\k\end{pmatrix}\cdot p^k\cdot q^{n-k}=\\ =\begin{pmatrix}5\\k\end{pmatrix}\cdot 0.6^k\cdot 0.4^{5-k}=\\ =\cfrac{5!}{k!\cdot(5-k)!}\cdot 0.6^k\cdot 0.4^{5-k}.$

The probability that team A wins 3 matches out of 5 is

$P(X=3)=\cfrac{5!}{3!\cdot2!}\cdot 0.6^{3}\cdot 0.4^{2}=\cfrac{4\cdot5}{2}\cdot0.216\cdot0.16=0.3456.$