one of the wagers in roulette is to bet that the ball will stop on a number that is a multiple of 3. (both 0 and 00 are not included) if the ball stops on such a number the player wins double the amount bet. if a player bets $1 compute the player’s expectation
To compute the player's expectation, we need to calculate the expected value of the outcome.
First, let's determine the probability of the ball stopping on a number that is a multiple of 3. On a roulette wheel, there are 38 numbers (1-36, 0, and 00). Out of these numbers, 12 are multiples of 3 (3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36). Therefore, the probability of the ball stopping on a multiple of 3 is 12/38.
If the player bets $1 and wins, they will receive double the amount bet, which is $2. If the player loses, they will lose the amount bet, which is $1.
Now, let's calculate the overall expectation:
(Probability of winning * Amount won) + (Probability of losing * Amount lost)
= (12/38 * $2) + (26/38 * -$1)
= ($24/38) - ($26/38)
= -$2/38
≈ -$0.0526
The player's expectation is approximately -$0.0526. This means, on average, the player can expect to lose about $0.0526 for every $1 bet on this wager.
To compute the player's expectation, we need to determine the probability of winning and losing, as well as the corresponding payouts.
In a roulette wheel, there are a total of 38 numbers, of which 12 are multiples of 3 (excluding 0 and 00). Therefore, the probability of the ball stopping on a multiple of 3 is:
Probability of winning = 12/38
If the player wins, they receive double the amount bet, which is $1 * 2 = $2.
The probability of losing is equal to the complement of the probability of winning:
Probability of losing = 1 - (12/38) = 26/38
When the player loses, they lose the amount of their bet, which is $1.
Now, we can calculate the player's expectation:
Expectation = (Probability of winning * Payout for winning) - (Probability of losing * Payout for losing)
= ((12/38) * $2) - ((26/38) * $1)
Let's calculate this:
Expectation = (12/38 * 2) - (26/38 * 1)
= 0.6316 - 0.6842
= -0.0526
Therefore, the player's expectation for this wager is -$0.0526 or -5.26 cents. This means that on average, the player is expected to lose 5.26 cents for every dollar they bet on this wager.