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Chapter 8: Problem 25

A red and a green die are rolled, and \(X\) is the sum of the numbers facing up.

Short Answer

Expert verified
The possible sums when rolling a red and green die are between 2 and 12, with the respective probabilities being: P(2)=1/36, P(3)=2/36, P(4)=3/36, P(5)=4/36, P(6)=5/36, P(7)=6/36, P(8)=5/36, P(9)=4/36, P(10)=3/36, P(11)=2/36, and P(12)=1/36.

Step by step solution

01

Identify Possible Outcomes for Each Die

Since these are standard six-sided dice, the possible outcomes for each die are the numbers from 1 to 6.
02

Determine the Sample Space for X

To find all the possible outcomes for the sum \(X\), we can use a table to list the sums for each combination of rolls for the red and green dice: | | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---| | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | 3 | 4 | 5 | 6 | 7 | 8 | 9 | | 4 | 5 | 6 | 7 | 8 | 9 |10 | | 5 | 6 | 7 | 8 | 9 |10 |11 | | 6 | 7 | 8 | 9 |10 |11 |12 | From this table, we can see that the minimum value for the sum is 2, and the maximum is 12.
03

Count the Frequency of Each Outcome

From the table, we can count how many times each sum appears in order to determine the frequency of each outcome: - 2: 1 time - 3: 2 times - 4: 3 times - 5: 4 times - 6: 5 times - 7: 6 times - 8: 5 times - 9: 4 times - 10: 3 times - 11: 2 times - 12: 1 time
04

Calculate the Probability of Each Outcome

There are a total of 36 possible outcomes in the sample space (6 outcomes for the red die multiplied by 6 outcomes for the green die). The probability for each sum can be calculated as the frequency of each outcome divided by the total number of possible outcomes: - P(2): 1/36 - P(3): 2/36 - P(4): 3/36 - P(5): 4/36 - P(6): 5/36 - P(7): 6/36 - P(8): 5/36 - P(9): 4/36 - P(10): 3/36 - P(11): 2/36 - P(12): 1/36
05

Display the Probability Distribution of X

Finally, we can display the probability distribution of the random variable \(X\): \(X\) | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | P(X) |1/36|2/36|3/36|4/36|5/36|6/36|5/36|4/36|3/36|2/36|1/36| The probability distribution table shows the probability of each possible sum when rolling a red and green die together.

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