Question

If you play a game that you can only either win or lose. The probability that you win any game is 35%. If you play the game 45 times, what is the probability that you win 15 if the 45 games, less than 15 games, between 5 and 9 games out of 45?

Answer 1

p=0.35

n=45

The probability that you win 15 if the 45 games

$P(X=x) = C^n_x p^x (1-p)^{n-x} \\ P(X=15) = C^{45}_{15} \times 0.35^{15} \times 0.65^{45-15} \\ = \frac{45!}{15!(45-15)!} \times 0.35^{15} \times 0.65^{30} \\ = 0.1219$

The probability that you win less than 15 games

$P(X<15) = \sum^{14}_{x=0} C^{45}_{x} \times 0.35^{x} \times 0.65^{45-x} = 0.3532$

The probability that you win between 5 and 9 games out of 45

$P(5<X<9) = \sum^{8}_{x=6} C^{45}_{x} \times 0.35^{x} \times 0.65^{45-x} = 0.0088$