Question

Roulette is a casino game that involves spinning a ball on a wheel that is marked with numbered squares that are red, black, or green. Half of the numbers 1-36 are red and half are black, 0 and 00 are green. Each number occurs only once on the wheel.

The most common bets are to bet on a single number or to bet on a color (red or black). The pocket in which the ball lands on the wheel determines the winning number and color. The ball can land on only one color and number at a time.

We begin by placing a bet on a number between 1 and 36. This bet pays 36 to 1 in most casinos, which means we will be paid $36 for each $1 we bet on the winning number. If we lose, we simply lose whatever amount of money we bet.

Calculate the probability that we will win on a single spin of the wheel.

Calculate the probability that we will lose.

What is the expected value of a bet on a single number if we bet $1?

What is the expected value of a bet on a single number if we bet $5?

What is the expected value of a bet on a single number if we bet $10?

Can you explain your responses to the three expected value questions?

We decide that we can certainly increase our chances of winning if we bet on a color instead of a number. This bet pays even money in most casinos. This means that for each dollar we bet, we will win $1 for choosing the winning color. So, if we bet $5 and win, we would keep our $5 and win $5 more. If we lose, we lose whatever amount of money we bet, just as before.

What is the probability that we will win on a single spin?

If we bet $60 on the winning color, will we win more or less than if we bet $8 on the winning number?

What is the expected value of a $1 bet on red?

What is the expected value of a $5 bet on red?

What is the expected value of a $10 bet on red?

How does the expected value of betting on a number compare to the expected value of betting on a color?

Are casinos really gambling when we place a bet against them? Explain.

Answer 1

If we bet $1 on a single spin ,Probability that what we will win on a single spin of the wheel=1/38

The sum of all probabilities is 1

So, the probability that we will lose = 1-1/38=37/38

Expected Value of a $1 bet on a single number:

Let y be your winnings resulting from a $1 bet on a single number; y has 2 possible values

y	-1	35
probability of getting y	37/38	1/38

E(y)= -1(37/38)+35(1/38)= -1/19= -0.523

So on average you win 5.3 cents on every such bet.

Similarly,Expected value of a bet on a single number if we bet $5:

y	-5	5*35=175
p(y)	37/38	1/38

E(y)= -5(37/38)+175(1/38)= -5/19= -0.26

Similarly,Expected value of a bet on a single number if we bet $10:

y	-10	10*35=350
p(y)	37/38	1/38

E(y)= -10(37/38)+350(1/38)= -10/19 = -0.53

As we have increased the amount of betting, there is a high chance of losing more as the expected value is getting negative.

Probability that we will win on a single spin if we bet on a colour is 18/38

If we bet $1 on red then:

y	-1	1
P(y)	20/38	18/38

E(y)= -1(20/38)+1(18/38)= -1/19 = -0.053

If we bet $5 on red then:

y	-5	5
P(y)	20/38	18/38

E(y)= -5(20/38)+5(18/38)= -1/19 = -0.26

If we bet $5 on red then:

y	-10	10
P(y)	20/38	18/38

E(y)= -10(20/38)+10(18/38)= -20/19 = -0.53

The expected values are cming out to be same in both the scenarios.

A casino is a facility which houses and accommodates certain types of gambling activities. The industry that deals in casinos is called the gaming industry. Casinos are most commonly built near or combined with hotels, restaurants, retail shopping, cruise ships or other tourist attractions.