A standard roulette wheel has numbers 1 through 36 alternately colored red and b

Question

A standard roulette wheel has numbers 1 through 36 alternately colored red and black. It also has a green 0 and a green 00 called “double zero.” The wheel is spun with a small white ball inside. When the wheel stops, the ball falls into a numbered slot, white determines the winners. Successive spins of a wheel yield independent results. There are many wagers you can place at the roulette table. For example, you may bet on the color, or on whether the number is odd or even, or on a single number. Note that, even though 0 is in fact an even number, in roulette both 0 and 00 count an neither odd nor even.

a) You are playing roulette, and you bet a dollar on the number 10. If 10 comes up you win $35(profit). If anything else comes up, you lose your dollar. What is your expected value for this bet? ______________

b) If you bet $1 on an even number, you win $1 (profit) if any of the even numbers 2 through 36 come up, and you lose your dollar if 0, 00, or any odd number between 1 and 35 comes up. What is your expected value for this wager? ________________

c) If you place a bet on the number 10 on the roulette wheel, the casino has the advantage. How much profit should I get from a $1 bet on number 10 to make the game fair? __________________

Explanation / Answer

a.

Profit(x)

Probability

x*Probability

35

1/38 = 0.0263

0.9211

-1

1-0.0263=0.9737

-0.9737

-0.0526

Expected value is -$0.05

The loss is $0.05

b.

Profit(x)

Probability

x*Probability

1

18/38 = 0.4737

0.4737

-1

1-0.4737=0.5263

-0.5263

-0.0526

Expected value is -$0.05

The loss is $0.05

c.

Profit(x)

Probability

x*Probability

1

0.0263

0.0263

-1

0.9737

-0.9737

-0.9474~-$0.95

Since it is loss of $0.95, you should make $0.95 for the game to be fair.

a.

Profit(x)

Probability

x*Probability

35

1/38 = 0.0263

0.9211

-1

1-0.0263=0.9737

-0.9737

-0.0526

Expected value is -$0.05

The loss is $0.05

b.

Profit(x)

Probability

x*Probability

1

18/38 = 0.4737

0.4737

-1

1-0.4737=0.5263

-0.5263

-0.0526

Expected value is -$0.05

The loss is $0.05

c.

Profit(x)

Probability

x*Probability

1

0.0263

0.0263

-1

0.9737

-0.9737

-0.9474~-$0.95

Since it is loss of $0.95, you should make $0.95 for the game to be fair.

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