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9 days ago

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g European roulette. The game of European roulette involves spinning a wheel with 37 slots: 18 red, 18 black, and 1 green. A bal

l 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

1 answer:

Leona [4.1K]9 days ago

6 0

Answer:

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

Step-by-step explanation:

In European roulette, a wheel with 37 slots is spun: 18 are red, 18 are black, and 1 is green.

Bettors can wager on either red or black. Winning on their chosen color means they double their bet, while losing means they forfeit their stake.

Define the probability for the ball landing in a red slot = $\frac{18}{37}$

The probability of it landing in a black slot = $\frac{18}{37}$

The probability for landing in the green slot = $\frac{1}{37}$

Since bets are only placed on red or black,

The winning probability = $\frac{18}{37}$

and losing probability = $\frac{18}{37}+\frac{1}{37}$

= $\frac{19}{37}$

If a gambler wins, they receive $3, and if they lose, it amounts to -$3.

The expected value of the total winnings is hence;

E(X) = $\sum X \times P(X)$

= $\$3 \times \frac{18}{37} + (-\$3 \times \frac{19}{37})$

= $\$3 \times (-\frac{1}{37})$ = -$0.081

Furthermore, the standard deviation of the total winnings is given by;

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

<pTherefore, $E(X^{2})=\sum X^{2} \times P(X)$

= $\$3^{2} \times \frac{18}{37} + (-\$3^{2} \times \frac{19}{37})$

= $\$9 \times (\frac{18}{37}+\frac{19}{37})$ = $9

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

= $\sqrt{8.993}$ = $2.99 ≈ $3

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

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