Question

What will be the probability that the sum of the two numbers that turn up is less than 11 when two dice are thrown simultaneously?

Answer 1

Find the sample space when two dice are thrown simultaneously. We have to find the probability of the pairs of two dice whose sum is less than 11. Using the sample space, find the favorable outcomes for sum less than 11.

Complete Step by Step Solution:
We are given in the question with the two dice and they are thrown simultaneously. Now, we have to find the total outcomes of the two dice when thrown simultaneously, it can also be said as sample space. There are 1, 2, 3, 4, 5 and 6 numbers in one dice. Since, there are two dice the sample space can be written as –
S = \left\\{ (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) \\\ \right\\}
Number of total outcomes or sample space, $n(S) = 36$
We have drawn the sample space when two dice are thrown simultaneously.
From the question, we know that we have to find the probability of the two numbers whose sum is less than 11. Now, from the above sample space we have to find those pairs of dice whose sum is less than 11. Therefore, the favorable outcomes whose sum is less than 11 are –
E = \left\\{ (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) \\\ (6,1)(6,2)(6,3)(6,4) \\\ \right\\}
Number of favorable outcomes when two dice are thrown simultaneously whose sum is less than 11, $n\left( E \right) = 33$
Now, we know that, probability is found by dividing the number of favorable outcomes by number of total outcomes –
$P\left( E \right) = \dfrac{{n\left( E \right)}}{{n\left( S \right)}}$
Putting the values in the above formula, we get –
$\Rightarrow P\left( E \right) = \dfrac{{33}}{{36}} \\\ \Rightarrow P\left( E \right) = \dfrac{{11}}{{12}} \\\$

Hence, $\dfrac{{11}}{{12}}$ is the required probability.

Note:
Total number of sample space can also be calculated by formula, ${m^n}$ , where, $m$ is the number of sides of dice or coin and $n$ is the number of coins or dice when thrown simultaneously. For example, two coins are thrown simultaneously, then, the number of sample space can be calculated as –
$\Rightarrow {2^2} = 4$ , as there are 2 sides in one – coin head and tail.