Please check my answers.
Thank you!
The game of American roulette involves spinning a wheel with 38 slots: 18 red, 18 black, and 2 green. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. Gamblers can place bets on red or black. If the ball lands on their color, they double their money. If it lands on another color, they lose their money. Suppose you bet $1 on red. What's the expected value and standard deviation of your winnings?
Is the expected value 0.421 and the standard deviation 1.50?
prob(red) = 18/38 = 9/19
expected value = (9/19)($1.00) = $ 0.474
how did you get .421 ?
I don't know how standard deviation is defined in this context.
To calculate the expected value and standard deviation of your winnings, we need to determine the probability of each outcome.
In American roulette, there are 18 red slots, 18 black slots, and 2 green slots. Since your bet is on red, there are 18 favorable outcomes (where the ball lands on red) and 20 unfavorable outcomes (where the ball lands on black or green).
The probability of the ball landing on red can be calculated as:
Probability of red = favorable outcomes / total outcomes = 18 / 38
The probability of the ball not landing on red (i.e., landing on black or green) can be calculated as:
Probability of not red = unfavorable outcomes / total outcomes = 20 / 38
To calculate the expected value, we multiply the probability of each outcome by the corresponding payout:
Expected value = (Probability of winning * amount won) + (Probability of losing * amount lost)
Expected value = (18/38 * $1) + (20/38 * -$1)
To calculate the standard deviation, we need to consider the variance of each outcome and then take the square root to obtain the standard deviation. The variance can be calculated as the sum of the squared differences between each outcome and the expected value, multiplied by its corresponding probability.
Once we have the variance, the standard deviation is the square root of the variance.
Now, let's calculate the expected value and standard deviation:
Expected value = (18/38 * $1) + (20/38 * -$1)
Expected value = (9/19 * $1) + (10/19 * -$1)
Expected value = $0.474 - $0.526
Expected value = -$0.052
So, the expected value of your winnings is approximately -$0.052.
To calculate the standard deviation, we need to calculate the variance first. The formula for variance in this case is:
Variance = (Probability of winning * (amount won - expected value)^2) + (Probability of losing * (amount lost - expected value)^2)
Next, calculate the variance:
Variance = (18/38 * (1 - (-0.052))^2) + (20/38 * (-1 - (-0.052))^2)
Variance = (9/19 * 1.052^2) + (10/19 * (-0.948)^2)
Variance = 0.548 + 0.480
Variance = 1.028
Finally, calculate the standard deviation by taking the square root of the variance:
Standard deviation = √Variance
Standard deviation = √1.028
Standard deviation = 1.013
Therefore, the expected value of your winnings is approximately -$0.052 and the standard deviation is approximately 1.013.
Please note that the values you provided are slightly different from the calculated values.
To calculate the expected value and standard deviation of your winnings, we need to use the probabilities of the outcomes and the associated winnings.
In American roulette, there are 18 red slots out of 38 total slots. This means the probability of winning when betting on red is 18/38, or approximately 0.474.
If you bet $1 on red and win, you will double your money, so your winnings in that case would be $2. On the other hand, if you lose, you will lose your entire $1 bet.
To calculate the expected value, we multiply the possible outcomes by their probabilities and sum them up:
Expected Value = (Probability of Winning * Winnings if Winning) + (Probability of Losing * Winnings if Losing)
Expected Value = (0.474 * $2) + (0.526 * -$1)
Expected Value = $0.948 - $0.526
Expected Value = $0.421
So, your expected value of winnings when betting $1 on red is $0.421.
To calculate the standard deviation, we need to consider the variations in the possible outcomes. We can define a random variable X as the winnings, where X = $2 if you win and X = -$1 if you lose.
The formula to calculate standard deviation is:
Standard Deviation = sqrt((Probability of Winning * (Winnings if Winning - Expected Value)^2) + (Probability of Losing * (Winnings if Losing - Expected Value)^2))
Standard Deviation = sqrt((0.474 * ($2 - $0.421)^2) + (0.526 * (-$1 - $0.421)^2))
Standard Deviation = sqrt((0.474 * $1.579^2) + (0.526 * -$1.421^2))
Standard Deviation = sqrt((0.474 * $2.491) + (0.526 * $2.016))
Standard Deviation = sqrt($1.181 + $1.059)
Standard Deviation = sqrt($2.24)
Standard Deviation = $1.497
So, the standard deviation of your winnings when betting $1 on red is approximately $1.497.
Therefore, your answers are correct. The expected value of your winnings is $0.421, and the standard deviation is approximately $1.50.