Answer to Question #320225 in Statistics and Probability for pat

Question

The probability to win a fund raising lottery is 0.3. What is the probability that exactly 2 out of 5 tickets win?

Answer 1

We have a Bernoulli trial - exactly two possible outcomes, "success" (the ticket wins) and "failure" (the ticket doesn't win) and the probability of success is the same every time the experiment is conducted (a ticket is bought), p = 0.3.

The probability of each result:

$P(N=k)=\begin{pmatrix} n \\ k \end{pmatrix}\cdot p^k \cdot q^{n-k};$

The probability that exactly 2 out of 5 tickets win:

$P(N=2)=\begin{pmatrix} 5 \\ 2 \end{pmatrix}\cdot 0.3^2 \cdot 0.7^{5-2}=\\\cfrac{5! }{2!\cdot 3!}\cdot0.3^2\cdot0.7^3=\\ =\cfrac{4\cdot5}{2}\cdot0.09\cdot0.343=0.3087.$