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) Expected amount of winnings = -$0.084

Standard deviation = $3.00

b) Expected amount of winnings = -$0.084

Standard deviation = $1.73

c) The expected winnings for both cases is exactly the same, but the spread of the wins about the expected value/mean is low in the case (b) compared to case (a).

So, with the expected winnings using the method in (a) and (b) the same, the risk associated with betting $3 dollar per game is higher because it has a higher standard deviation. This points to higher spread of losses and wins betting $3 once than betting $1 three different times.

Step-by-step explanation:

(a) Suppose you play roulette and bet $3 on a single round. What is the expected value and standard deviation of your total winnings?

It's either one wins or not with a bet of $3.

We can then obtain the probability mass function.

Random variable X represents the amount of winnings.

- If one bets $3 and picks the right colour, the person wins $6.

X = amount of winnings = 6 - 3 = $3

probability of winning = (18/37) = 0.486

- If one bets $3 and picks the wrong colour, then the person wins $0.

X = amount of winnings = 0 - 3 = -$3

Probability of losing = (19/37) = 0.514

The probability mass function is then

X | P(X)

3 | 0.486

-3 | 0.514

E(X) = Σ xᵢpᵢ

xᵢ = each variable

pᵢ = probability of each variable

E(X) = (3×0.486) + (-3×0.514) = -$0.084

Standard deviation = √(variance)

Variance = Var(X) = Σx²p − μ²

μ = E(X) = -0.084

Σx²p = (3²×0.486) + [(-3)² × 0.514] = 9

Variance = 9 - (-0.084)² = 8.993

Standard deviation = √8.993 = 2.999 = $3

(b) Suppose you bet $1 in three different rounds. What is the expected value and standard deviation of your total winnings?

With a bet of $1,

It's either one wins or not with a bet of $1.

We can then obtain the probability mass function.

Random variable X represents the amount of winnings.

- If one bets $1 and picks the right colour, the person wins $2.

X = amount of winnings = 2 - 1 = $1

probability of winning = (18/37) = 0.486

- If one bets $1 and picks the wrong colour, then the person wins $0.

X = amount of winnings = 0 - 1 = -$1

Probability of losing = (19/37) = 0.514

The probability mass function is then

X | P(X)

1 | 0.486

-1 | 0.514

E(X) = Σ xᵢpᵢ

xᵢ = each variable

pᵢ = probability of each variable

E(X) = (1×0.486) + (-1×0.514) = -$0.028

For 3 rounds of $1, the expected amount of winnings = 3 × -0.028 = -$0.084

Variance = Var(X) = Σx²p − μ²

μ = E(X) = -0.028

Σx²p = (1²×0.486) + [(-1)² × 0.514] = 1

Variance = 1 - (-0.028)² = 0.999216

Variance for combining 3 independent distributions = Σ λᵢ²σᵢ²

Variance of 3 rounds of $1 bets = 3[1²× var(X)] = 3 × var(X) = 3 × 0.999216 = 2.997648

Standard deviation = √(variance) = √(2.997648) = $1.73

(c) How do your answers to parts (a) and (b) compare? What does this say about the riskiness of the two games?

The expected winnings for both cases is exactly the same, but the spread of the wins about the expected value/mean is low in the case (b) compared to case (a).

So, with the expected winnings using the method in (a) and (b) the same, the risk associated with betting $3 dollar per game is higher because it has a higher standard deviation. This points to higher spread of losses and wins betting $3 once than betting $1 three different times.

Hope this Helps!!!

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step-by-step explanation:

the central idea is that:

individuals should refuse to cooperate with institutions they believe unjust.

thoreau felt  that governments were more often agents of corruption than justice, even democracies, and that just because the majority feels a certain way does not make it just or correct in absolute terms. he uses the fact that slavery exists in our democracy to support this idea.

he goes on to say that, as a result, people should live true to their own internal laws regarding right and wrong, as a product of reflection and a moral life, rather than following every law that society imposes. 

step-by-step explanation: