Question

On a roulette wheel, the probability of "red" is 18/38, In 10 spins what is the probability of exactly 8 red outcomes?

Answer 1

We have a Bernoulli trial - exactly two possible outcomes, "success" (red came out) and "failure" (red didn't come out) and the probability of success is the same every time the experiment is conducted (the roulette wheel spinned), $p=\cfrac{18}{38}=\cfrac{9}{19}, q=1-\cfrac{9}{19}=\cfrac{10}{19}, n=10.$

The probability of exactly k red outcomes

$P(X=k)=\begin{pmatrix}n\\k\end{pmatrix}\cdot p^k\cdot q^{n-k}=\\ =\begin{pmatrix}10\\k\end{pmatrix}\cdot\begin{pmatrix}\cfrac{9}{19}\end{pmatrix} ^k\cdot \begin{pmatrix}\cfrac{10}{19}\end{pmatrix}^{10-k}=\\ =\cfrac{10!}{k!\cdot(10-k)!}\cdot \begin{pmatrix}\cfrac{9}{19}\end{pmatrix}^k\cdot \begin{pmatrix}\cfrac{10}{19}\end{pmatrix}^{10-k}.$

The probability of exactly 8 red outcomes

$P(X=8)=\cfrac{10!}{8!\cdot2!}\cdot \begin{pmatrix}\cfrac{9}{19}\end{pmatrix}^{8}\cdot \begin{pmatrix}\cfrac{10}{19}\end{pmatrix}^{2}=0.03159.$