Question

consider the following dice game one dollar for rolling a 1, 2,or 3; 5 dollars for rolling a 4 or 5; 50 dollars for rolling a 6. find the expected value and standard deviation for the game

Answer 1

The expected value of the game is $10.50 and the standard deviation is 25.08.

Step-by-step explanation:

To find the expected value and standard deviation of the dice game, we need to determine the probabilities for each outcome and multiply them by their respective payouts.

Let X be the random variable representing the winnings.

• If a 1, 2, or 3 is rolled, the probability is 3/6, and the payout is $1. So, the expected value for this outcome is (3/6) * (1) = $0.50.
• If a 4 or 5 is rolled, the probability is 2/6, and the payout is $5. So, the expected value for this outcome is (2/6) * (5) = $1.67.
• If a 6 is rolled, the probability is 1/6, and the payout is $50. So, the expected value for this outcome is (1/6) * (50) = $8.33.

To find the overall expected value, we sum the expected values of each outcome: $0.50 + $1.67 + $8.33 = $10.50.

To find the standard deviation, we need to calculate the variance first. The variance is calculated by summing the squared differences between each outcome and the expected value, multiplied by their respective probabilities. Then, take the square root of the variance to find the standard deviation.

Calculating the variance: (3/6 * (1 - 10.50)^2) + (2/6 * (5 - 10.50)^2) + (1/6 * (50 - 10.50)^2) = 628.92.

Therefore, the standard deviation is the square root of the variance: √628.92 = 25.08.