A black and a red die are rolled simultaneously. The probability of obtaining a sum greater than 9, given that the black die resulted in a 5 is

The correct option/answer is 1/3.

Let us break down the solution step-by-step using conditional probability.

Step 1: Understand the Given Condition
We are given that a black die and a red die are rolled. We are told that the black die resulted in a 5.
Let B be the event that the black die shows a 5. Since a single fair die has 6 possible outcomes, and we are looking at the roll of the red die alongside this fixed outcome, let's write out the possible outcomes for the pair (Black, Red):
$$S=\left\{(5,1),,,(5,2),,,(5,3),,,(5,4),,,(5,5),,,(5,6)\right\}$$
This gives us a total of 6 possible outcomes in our restricted sample space. Thus, the number of elements in the sample space under this condition is:
$n\left(B\right)=6$

Step 2: Identify the Favorable Outcomes
We want to find the probability that the sum of the numbers on the two dice is greater than 9.
Let us check the sum for each of our possible outcomes:

• For (5, 1), the sum is 5 + 1 = 6 (not greater than 9)
• For (5, 2), the sum is 5 + 2 = 7 (not greater than 9)
• For (5, 3), the sum is 5 + 3 = 8 (not greater than 9)
• For (5, 4), the sum is 5 + 4 = 9 (not greater than 9, as it is exactly 9)
• For (5, 5), the sum is 5 + 5 = 10 (greater than 9)
• For (5, 6), the sum is 5 + 6 = 11 (greater than 9)

Step 3: Calculate the Conditional Probability
The required conditional probability is the ratio of the number of favorable outcomes to the total number of conditional outcomes:
$$P=\frac{n\left(\text{Favorable}\right)}{n\left(B\right)}=\frac{2}{6}=\frac{1}{3}$$
This confirms that the probability of obtaining a sum greater than 9, given that the black die resulted in a 5, is indeed 1/3.