Question

g European roulette. The game of European roulette involves spinning a wheel with 37 slots: 18 red, 18 black, and 1 green. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. Gamblers can place bets on red or black. If the ball lands on their color, they double their money. If it lands on another color, they lose their money. (a) Suppose you play roulette and bet $3 on a single round. What is the expected value and standard deviation of your total winnings

Answer 1

The expected value and standard deviation of your total winnings are -$0.081 and $3 respectively.

Explanation:

We are given that the game of European roulette involves spinning a wheel with 37 slots: 18 red, 18 black, and 1 green.

Gamblers can place bets on red or black. If the ball lands on their color, they double their money. If it lands on another color, they lose their money.

Let the probability of the ball landing on red slot =
`(18)/(37)`

The probability of the ball landing on black slot =
`(18)/(37)`

The probability of the ball landing on green slot =
`(1)/(37)`

Now, it is stated that Gambler can place bets only on the red or black slot, so;

The probability of winning the bet will be =
`(18)/(37)`

and the probability of losing the bet will be =
`(18)/(37)+(1)/(37)`

=
`(19)/(37)`

If the gambler wins he gets $3 and if he loses he will get -$3.

So, the expected value of gambler's total winnings is given by;

E(X) =
`\sum X * P(X)`

=
`\$3 * (18)/(37) + (-\$3 * (19)/(37))`

=
`\$3 * (-(1)/(37))` = -$0.081

Now, the standard deviation of gambler's total winnings is given by;

S.D.(X) =
`\sqrt{(\sum X^(2) * P(X))-(\sum X * P(X))^(2) }`

So,
`E(X^(2))=\sum X^(2) * P(X)`

=
`\$3^(2) * (18)/(37) + (-\$3^(2) * (19)/(37))`

=
`\$9 * ((18)/(37)+(19)/(37))` = $9

Now, S.D.(X) =
`\sqrt{\$9-(-\$0.081)^(2) }`

=
`√(8.993)` = $2.99 ≈ $3

Hence, the expected value and standard deviation of your total winnings are -$0.081 and $3 respectively.