Question

A and B throw a pair of dice respectively. If A throws dice such that the total of the numbers on the dice is 9, find B's chance of getting a higher number than A.

Answer 1

Hint: In this question, we are given that A and B throw a pair of dice respectively and A throws dice such that the sum of numbers is 9. We have to find B's chances of getting a higher number than A. Hence, we have to find a probability of getting a sum greater than 9 that is 10, 11 or 12. For this, we will first create sample space which will give us total events of when two dice are thrown. After that we will find favorable outcomes out of total events and count them to find the number of favorable outcomes. At last, we will find probability using the formula:
$\text{Probability}=\dfrac{\text{Number of favorable outcomes}}{\text{Number of total outcomes}}$

Complete step by step answer:
Here, A threw a pair of dice to get sum as 9. We have to find the probability that B gets a sum greater than 9. Hence, we have to find the probability that B gets a sum of 10, 11 or 12.
Let us first draw sample space of possibilities. Since, every dice has 6 numbers, so total possibilities will be $6\times 6=36$.
Therefore, sample space becomes:
(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)
Hence, the total number of outcomes is 36.
With the help of this sample space, we will find required elements.
Since, we need pairs with sums greater than 9. So let us look out for numbers with sums equal to 10, 11 or 12. Pair of numbers having sum 10 are:
(4,6), (6,4), (5,5).
Pair of numbers having sum 11 are:
(5,6), (6,5).
Pair of numbers having sum 12 are:
(6,6).
As we can see, the total number of possibilities are 3+2+1 = 6.
Hence, the number of favorable outcomes = 6.
As found earlier, total number of outcomes = 36.
Now, $\text{Probability}=\dfrac{\text{Number of favorable outcomes}}{\text{Number of total outcomes}}$ therefore,
$\text{Probability}=\dfrac{\text{6}}{\text{36}}$
Simplifying we get:
$\text{Required probability}=\dfrac{1}{6}$

Hence, chances of B's getting a number higher than A while rolling two dice are $\dfrac{1}{6}$.

Note: Students should carefully count all the possibilities while determining a favorable outcome. Try to understand the problem and then find what numbers will be covered under favorable outcomes. Sample space should be made carefully without repetition.