ex4. a card is to be selected from an ordinary - deck of 52 cards. suppose that a casino will pay $10 if you…

Explanation:

Step1: Calculate probability of selecting an ace

There are 4 aces in a 52 - card deck. So the probability of selecting an ace, $P(A)=\frac{4}{52}=\frac{1}{13}$.

Step2: Calculate probability of not selecting an ace

The probability of not selecting an ace, $P(\overline{A}) = 1 - P(A)=1-\frac{1}{13}=\frac{12}{13}$.

Step3: Calculate the casino's expected gain

If an ace is selected, the casino pays $10, so its gain is - 10. If an ace is not selected, the player pays 1, so the casino's gain is 1. The expected value formula is $E = x_1P(x_1)+x_2P(x_2)$. Here, $x_1=- 10$, $P(x_1)=\frac{1}{13}$, $x_2 = 1$, $P(x_2)=\frac{12}{13}$. Then $E=-10\times\frac{1}{13}+1\times\frac{12}{13}=\frac{-10 + 12}{13}=\frac{2}{13}\approx0.15$.

Answer:

The casino expects to win approximately $$0.15$ per game.