Question

6. P(x ≤80) C. If each pair of die is "loaded" so that one comes up half as often as it should, six comes up half again as often as it should, and the probabilities of the other faces are unaltered, then the probability distribution for the sum X of the number of dots on the top faces when the two are rolled is 7 6 5 9 8 11 10 3 4 12 2 24 20 4 16 22 12 8 9 16 12 1 P(x) 144 144 144 144 144 144 144 144 144 144 144 X Compute each of the following. 7. P(5≤X ≤9) 8. P(X≥7)​

Answer 1

very long equation

Explanation:

To compute the probabilities requested, we can use the probability distribution provided:

| X | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

|------|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|

| P(X) | 1/24 | 1/8 | 1/16 | 5/48 | 7/48 | 1/4 | 1/16 | 5/48 | 1/8 | 5/48 | 1/24 |

7. P(5 ≤ X ≤ 9)

We need to find the probability that the sum of the dice is between 5 and 9 (inclusive), which means X can take the values of 5, 6, 7, 8, or 9. We add the probabilities for these values to get:

P(5 ≤ X ≤ 9) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9)

= 5/48 + 7/48 + 1/4 + 1/16 + 5/48

= 0.40625

Therefore, the probability that the sum of the dice is between 5 and 9 (inclusive) is 0.40625.

8. P(X ≥ 7)

We need to find the probability that the sum of the dice is greater than or equal to 7, which means X can take the values of 7, 8, 9, 10, 11, or 12. We add the probabilities for these values to get:

P(X ≥ 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)

= 1/4 + 1/16 + 5/48 + 1/8 + 5/48 + 1/24

= 0.65625

Therefore, the probability that the sum of the dice is greater than or equal to 7 is 0.65625.

To find the mean of X, we can use the formula:

mean = Σ(X * P(X))

where Σ represents the sum of the values over all possible outcomes.

So, we have:

mean = (7*144 + 6*144 + 5*144 + 9*144 + 8*144 + 11*144 + 10*144 + 3*144 + 4*144 + 12*144 + 2*144 + 24*144 + 20*144 + 4*144 + 16*144 + 22*144 + 12*144 + 8*144 + 9*144 + 16*144 + 12*144) / 144^2

mean = 1980/144

mean = 13.75

To find the standard deviation, we can use the formula:

σ = sqrt[Σ(X - mean)^2 * P(X)]

So, we have:

σ = sqrt[(7-13.75)^2*144 + (6-13.75)^2*144 + (5-13.75)^2*144 + (9-13.75)^2*144 + (8-13.75)^2*144 + (11-13.75)^2*144 + (10-13.75)^2*144 + (3-13.75)^2*144 + (4-13.75)^2*144 + (12-13.75)^2*144 + (2-13.75)^2*144 + (24-13.75)^2*144 + (20-13.75)^2*144 + (4-13.75)^2*144 + (16-13.75)^2*144 + (22-13.75)^2*144 + (12-13.75)^2*144 + (8-13.75)^2*144 + (9-13.75)^2*144 + (16-13.75)^2*144 + (12-13.75)^2*144] / 144^2

σ = sqrt[3780.9375] / 12

σ = 5.8125 / 12

σ = 0.4844 (rounded to four decimal places)

Therefore, the mean of X is 13.75 and the standard deviation is 0.4844.