A black and a red dice are rolled. (a) Find the conditional probability of obtaining a sum greater than $9$, given that the black die resulted in a $5$. (b) Find the conditional probability of obtaining the sum $8$, given that the red die resulted in a number less than $4$.

For the first part, the sample space of rolling a 5 on the black die has 6 outcomes, and getting a sum greater than 9 requires the red die to be 5 or 6, giving a probability of 2/6 or 0.33. For the second part, the red die resulting in a number less than 4 gives a sample space of 18 outcomes. To obtain a sum of 8, the possible pairs are (2, 6), (3, 5), (5, 3) and (6, 2), which is 4 outcomes, yielding a probability of 4/36 or 0.11. The values are 0.33 and 0.11.

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