Question

In European roulette, the wheel is divided into 37 compartments numbered 1 through 36 and 0. (In American roulette there are 38 compartments numbered 1 through 36, 0, and 00.) One-half of the numbers 1 through 36 are red, the other half are black, and the number 0 is green. Find the expected value of the winnings on a $8 bet placed on red in European roulette. (Round your answer to three decimal places.)

Answer 1

The expectation is -$0.216.

Explanation:

Consider the provided information.

In European roulette, the wheel is divided into 37 compartments numbered 1 through 36 and 0.

One-half of the numbers 1 through 36 are red, the other half are black, and the number 0 is green.

We need to find the expected value of the winnings on a $8 bet placed on red in European roulette.

Here the half of 36 is 18.

That means 18 compartments are red and 18 are black.

The probability of getting red in European roulette is 18/37

The probability of not getting red in European roulette is 19/37. Because 18 are black and 1 is green.

If the ball lands on a red number, the player wins the amount of his bet.

The favorable outcomes are 18/37 and unfavorable are 19/37.

Let S be possible numerical outcomes of an experiment and P(S) be the probability.

The expectation can be calculated as:

E(x) = sum of S × P(S)

For
`S_1 = 8`

`P(S_1) = (18)/(37)`

For
`S_2 = -8`(negative sign represents the loss)

`P(S_2) = (19)/(37)`

Now, use the above formula.

`E(x) = 8* (18)/(37)-8* (19)/(37)\\E(x) =-0.216`

Hence, the expectation is -$0.216.