Probability of Sum Greater Than 9

This problem involves finding the probability of an event occurring, given that another specific event has already occurred. This is known as conditional probability.

Dice Roll Events Defined

We are rolling two dice, one black and one red, simultaneously. Let's define the events involved:

• Event A: The sum of the numbers shown on the two dice is greater than 9.
• Event C: The black die shows the number 5.

We are asked to find the probability of Event A occurring, given that Event C has occurred. This is represented as $P(A|C)$.

Black Die Condition Analysis

The condition given is that the black die resulted in a 5. This means we only need to consider the outcomes where the black die is 5. The possible outcomes are:

There are 6 possible outcomes when the black die shows a 5, as the red die can take any value from 1 to 6. This set of 6 outcomes forms our reduced sample space for this conditional probability problem.

Sum Greater Than 9 Outcomes

Within this reduced sample space (where the black die is 5), we need to identify the outcomes where the sum of the numbers on both dice is greater than 9.

• Outcome (5, 1): Sum = $5 + 1 = 6$. This is not greater than 9.
• Outcome (5, 2): Sum = $5 + 2 = 7$. This is not greater than 9.
• Outcome (5, 3): Sum = $5 + 3 = 8$. This is not greater than 9.
• Outcome (5, 4): Sum = $5 + 4 = 9$. This is not greater than 9.
• Outcome (5, 5): Sum = $5 + 5 = 10$. This is greater than 9.
• Outcome (5, 6): Sum = $5 + 6 = 11$. This is greater than 9.

The outcomes that satisfy the condition (black die is 5) and the event (sum > 9) are (5, 5) and (5, 6). Therefore, there are 2 favorable outcomes.

Calculating Conditional Probability

The conditional probability $P(A|C)$ is calculated by dividing the number of outcomes that satisfy both Event A and Event C by the number of outcomes that satisfy Event C.

$ P(\text{Sum} > 9 \mid \text{Black Die} = 5) = \frac{\text{Number of outcomes where Sum} > 9 \text{ and Black Die} = 5}{\text{Number of outcomes where Black Die} = 5} $

From our analysis:

• The number of outcomes where the sum is greater than 9 and the black die is 5 is 2.
• The total number of outcomes where the black die is 5 is 6.

So, the probability is:

$ P(\text{Sum} > 9 \mid \text{Black Die} = 5) = \frac{2}{6} $

Simplifying this fraction, we get:

$ \frac{2}{6} = \frac{1}{3} $

Therefore, the probability of obtaining a sum greater than 9, given that the black die resulted in a 5, is $1/3$.