Question

A and B throw with 3 dice: if A throws 8, what is B′s chance of throwing a higher number ?

Answer 1

Hint : To answer the chance of getting a higher number by B, first we will calculate the total number of possible cases then calculate the number of favorable cases for getting a higher number by B. thereafter use the formula of probability and find the chance of B to get a higher number.

Complete step by step solution:
Given in the question that A throws 8.
We have to calculate the chance of getting higher number than 8 by B
We know that when a dice is thrown there are a total 6 sample spaces, when two dice are thrown the total number of sample spaces is 36 while there are 216 total sample spaces when three dice are thrown.
We calculated above results by the formula that
Total number of sample space = ${6^n}$ where n is number of dices
We would find the cases where B throws a number less than or equal to 8 and then we subtract it from 216.
So the number of cases in which B get less than 8 is as follows
Number of cases when sum of 3 is on the dice is
$\left( {1,1,1} \right) = 1$
Number of cases when sum of 4 is on the dice is
$(1,1,2);(1,2,1);(2,1,1) = 1 + 2 = 3$
Number of cases when sum of 5 is on the dice is
$\left( {1,2,2} \right);\left( {2,1,2} \right);\left( {2,2,1} \right);\left( {3,1,1} \right);\left( {1,1,3} \right);\left( {1,3,1} \right) = 3 + 3 = 6$
Number of cases when sum of 6 is on the dice is $6 + 4 = 10$
Number of cases when sum of 7 is on the dice is $10 + 5 = 15$
Hence, total no of cases of B throwing higher = total number of cases – number of cases in which B gets less than 8 is $216 - \left( {1 + 3 + 6 + 10 + 15 + 21} \right) = 160$
Now, we know that the probability is the ratio of the number of favorable cases and the total number of cases.
So, probability is given as:

$$P = \dfrac{{160}}{{216}} \\\ \Rightarrow P = \dfrac{{20}}{{27}} \\\ \Rightarrow P = 0.74 \;$$

So, the correct answer is “ P = 0.74”.

Note : Students also can calculate the probability by counting the total number of cases in which B gets higher than 8. But don’t go for that because it is quite difficult to count so it is better to count unfavorable cases then subtract it from the total number of cases such that we will get a total number of favorable cases.