Questions: Let A be the event of rolling a sum of 8, and let B be the event of rolling doubles. The sample space consists of all possible rolls of two dice. Since there are 6 ways to roll the first 6-sided die, and 6 ways to roll the second 6-sided die, there are 6 × 6 = elements in the sample space. Recall the addition rule of probabilities: P(A or B) = P(A) + P(B) - P(A and B) Thus, we need to find all of the values on the right hand side of that equation.

Let A be the event of rolling a sum of 8, and let B be the event of rolling doubles. The sample space consists of all possible rolls of two dice. Since there are 6 ways to roll the first 6-sided die, and 6 ways to roll the second 6-sided die, there are 6 × 6 = elements in the sample space.

Recall the addition rule of probabilities:
P(A or B) = P(A) + P(B) - P(A and B)

Thus, we need to find all of the values on the right hand side of that equation.

Transcript text: Let $A$ be the event of rolling a sum of 8 , and let $B$ be the event of rolling doubles. The sample space consists of all possible rolls of two dice. Since there are 6 ways to roll the first 6-sided die, and 6 ways to roll the second 6 -sided die, there are $6 \times 6=$ $\square$ elements in the sample space. Recall the addition rule of probabilities: \[ P(A \text { or } B)=P(A)+P(B)-P(A \text { and } B) \] Thus, we need to find all of the values on the right hand side of that equation.

Solution

Solution Steps

To solve this problem, we need to calculate the probabilities of events $ A $, $ B $, and $ A \text{ and } B $ in the context of rolling two six-sided dice. First, determine the total number of outcomes in the sample space, which is $ 6 \times 6 = 36 $. Then, identify the outcomes that result in a sum of 8 (event $ A $) and the outcomes that result in doubles (event $ B $). Finally, find the outcomes that satisfy both conditions (event $ A \text{ and } B $) and use the addition rule of probabilities to find $ P(A \text{ or } B) $.

Step 1: Total Outcomes

The total number of outcomes when rolling two six-sided dice is calculated as: \[ \text{Total Outcomes} = 6 \times 6 = 36 \]

Step 2: Event A - Sum of 8

The outcomes that result in a sum of 8 are: \[ \text{Event A} = \{(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)\} \] The number of outcomes in Event A is $5$.

Step 3: Event B - Doubles

The outcomes that result in doubles are: \[ \text{Event B} = \{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)\} \] The number of outcomes in Event B is $6$.

Step 4: Event A and B - Sum of 8 and Doubles

The outcomes that satisfy both conditions (sum of 8 and doubles) are: \[ \text{Event A and B} = \{(4, 4)\} \] The number of outcomes in Event A and B is $1$.

Step 5: Calculate Probabilities

Using the counts from the previous steps, we can calculate the probabilities: \[ P(A) = \frac{5}{36} \approx 0.1389 \] \[ P(B) = \frac{6}{36} = \frac{1}{6} \approx 0.1667 \] \[ P(A \text{ and } B) = \frac{1}{36} \approx 0.0278 \]

Step 6: Addition Rule of Probabilities

Using the addition rule: \[ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \] Substituting the values: \[ P(A \text{ or } B) = \frac{5}{36} + \frac{6}{36} - \frac{1}{36} = \frac{10}{36} = \frac{5}{18} \approx 0.2778 \]