The probability of winning a match for team A is 0.6. Find the probability of winning 3 matches out of 5.
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}=\\\\\n=\\begin{pmatrix}5\\\\k\\end{pmatrix}\\cdot 0.6^k\\cdot 0.4^{5-k}=\\\\\n=\\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."
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