Question

2 and 3

Suomit Anemer Save Progress 2. + ถฯ points Previous Answes AufExc412e.oss at00 oe of the wager, in the game of roulette is to place . bet that the bal will land on . red number (eighteen of the number, sre b ack, as re red, md two-e gmm.) , ehe bil lanean. number, the player wins the amount of his bet. ir a payer bets 4. nnd the player's expectation. (Round your ant er to deonal place.) Need Help? My Notes Ask Tour 140 Submissions utiple of 3 (Both 0 and 00 are not ineluded ) If the ball stops on such a number, the player wins double the One of the wagers in roulette is to bet that the ball will stop on a number that is a rm amount bet. ir a player bets s2, compute the player's expectation. (Round your answer to two decimal places) Need Help?

Answer 1

2.

One of the wagers in the game of roulette is to place a bet that the ball will land on a red number (18 of the numbers are black, 18 are red and 2 are green). If the ball lands on a red number, the player wins the amount of his bet. If a player bets $4, find the player’s expectation.

A roulette wheel in the U.S. contains 38 equally sized spaces. The wheel is spun and a ball randomly lands in one of these spaces. Two spaces are green and have numbers 0 and 00 on them. The other spaces are numbered from 1 to 36. Half of these remaining spaces are red and half of them are black. Different wagers can be made on where the ball will end up landing. A common bet is to choose a color, such as red, and wager that the ball will land on any of the 18 red spaces.

Probabilities for Roulette

Since the spaces are the same size, the ball is equally likely to land in any of the spaces. This means that a roulette wheel involves a uniform probability distribution. The probabilities that we will need to calculate our expected value are as follows:

There are a total of 38 spaces, and so the probability that a ball lands on one particular space is 1/38.

There are 18 red spaces, and so the probability that red occurs is 18/38.

There are 20 spaces that are black or green, and so the probability that red does not occur is 20/38.

We use the above information with the formula for expected value. Since we have a discrete random variable X for net winnings, the expected value of betting $1 on red in roulette is

P (Red) x (Value of X for Red) + P (Not Red) x (Value of X for Not Red) = 18/38 x 4 + 20/38 x (-4) = 1.89473684 - 2.10526316 = -0.21052632

3. One of the wagers in the game of roulette is to place a bet that the ball will land on a number that is a multiple of 3(both 0 and 00 are not included) (18 of the numbers are black, 18 are red and 2 are green). If the ball lands on such a number, the player wins double the amount of his bet. If a player bets $2, find the player’s expectation.

Multiples of 3 between 1 to 36 are 3,6,9,12,15,18,21,24,27,30,33,36

Number of multiples of 3 between 1 to 36 are 12.

Therefore, P (multiple of 3) = 12/38

The expected value of betting $2(If the ball lands on such a number, the player wins double the amount of his bet) on a multiple of 3 is =

P (multiple of 3) x (Value of X for multiple of 3) + P (Not a multiple of 3) x (Value of X for Not multiple of 3)

(12/38)*4 + (26/38) * (-2) = 1.2631579 - 1.36842105 = - 0.10526315