The game of European roulette slots: 18 red, 18 black, and I green. A ball is spun onto the wheel and will eventually land ina 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, thev pean roulette. The game of European roulette 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? deviation of your total winnings? of the two games? (b) Suppose you bet $1 in three different rounds. What is the expected value and standard (c) How do your answers to parts (a) and (b) compare? What does this say about the riskiness

(a) You bet $3 on a single round which means that if you win the game, your amount will double ($6), your profit will be $3. Whereas, if you lose the round, your profit will be -$3. You can only bet on red or black and both have 18 slots each.

So, the probability of landing the ball in a red/black slot = 18/37. This is the probability of winning. The probability of losing can be calculated as 1-18/37 = 19/37.

We can make a probability distribution table:

x 3 -3

P(X=x) 18/37 19/37

Expected value E(x) can be calculated as:

E(x) = ∑ x.P(x)

= (3)(18/37) + (-3)(19/37)

E(x) = -0.081

Standard deviation can be calculated by the following formula:

Var(x) = E(x²) - E(x)²

S.D = √Var(x)

We need to first calculate E(x²).

E(x²) = ∑x².P(x)

= (3)²(18/37) + (-3)²(19/37)

= (9)(18/37) + (9)(19/37)

E(x²) = 9

Var(x) = E(x²) - E(x)²

= 9 - (-0.081)²

Var(x) = 8.993

S.D = √8.993

S.D = 2.99 ≅ 3

(b) Now, the betting price is $1 and 3 rounds are played. We will compute the expectation for one round and then add it thrice to find the expectation for three rounds. Similarly, for the standard deviations, we will add the individual variances and then consider the square root of it.

E(x) = ∑ x.P(x)

= (1)(18/37) + (-1)(19/37)

E(x) = -0.027

Standard deviation can be calculated by the following formula:

Var(x) = E(x²) - E(x)²

S.D = √Var(x)

We need to first calculate E(x²).

E(x²) = ∑x².P(x)

= (1)²(18/37) + (-1)²(19/37)

= (1)(18/37) + (1)(19/37)

E(x²) = 1

Var(x) = E(x²) - E(x)²

= 1 - (-0.027)²

Var(x) = 0.9992

The expectation for one round is -0.027

For three rounds,

E(x₁ + x₂ + x₃) = E(x₁) + E(x₂) + E(x₃)

= (-0.027) + (-0.027) + (-0.027)

E(x₁ + x₂ + x₃) = -0.081

Similarly, the variance for one round is 0.9992.

Var (x₁ + x₂ + x₃) = Var(x₁) + Var(x₂) + Var(x₃)

= 0.9992 + 0.9992 + 0.9992

Var (x₁ + x₂ + x₃) = 2.9976

S.D = √2.9976

S.D = 1.73

(c) The expected values for both part (a) and (b) are the same but the standard deviation is lower in part (c) as compared to (b). Since the standard deviation is less in part (c), it means that it is less risky to bet $1 in three different rounds as compared to betting $3 in a single round.