Question

A pair of dice is rolled together till a sum of either 5 or 7 is obtained. Then the probability that 5 comes before 7 is $\dfrac{k}{10}$. Find the value k.

Answer 1

Hint: Find the number of combinations for a sum of 5. Similarly, find the number of combinations for a sum of 7. Add them to find the total combinations. Divide the number of combinations for a sum of 5 by total number of combinations to get the final answer.

Complete step-by-step answer:
In this question, we are given that a pair of dice is rolled together till a sum of either 5 or 7 is obtained. Then the probability that 5 comes before 7 is $\dfrac{k}{10}$.
We need to find the value of k.
When we roll a pair of dice together, there are 11 possibilities of sum: from 2 to 12, and total 36 different outcomes ranging from (1, 1) to (6, 6).
In this question, we are concerned only with the sum of either 5 or 7.
For a sum of 5, we need to have one of the following combinations on the face of the pair of dice: (1, 4), (2, 3), (3, 2), (4, 1).
So, there are 4 outcomes which have the sum of 5.
For a sum of 7, we need to have one of the following combinations on the face of the pair of dice: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1).
So, there are 6 outcomes which have the sum 7.
So, in total we have 10 outcomes we are concerned with out of 4 outcomes are favourable and 6 outcomes are non favourable.
So, the probability that 5 comes before 7 is $\dfrac{4}{10}$
Hence, the value of k is 4.

Note: In this question, it is important to know that when we roll a pair of dice together, there are 11 possibilities of sum: from 2 to 12, and total 36 different outcomes ranging from (1, 1) to (6, 6). Also, realise that in total we have 10 outcomes we are concerned with out of 4 outcomes are favourable and 6 outcomes are non favourable.