Two dice, one blue and one grey, are thrown at the same time. Write down all the possible outcomes. What is the probability that the sum of the two numbers appearing on the top of the dice is 8 ?

# Two dice, one blue and one grey, are thrown at the same time. Write down all the possible outcomes. What is the probability that the sum of the two numbers appearing on the top of the dice is 8 ?

1. A
$\frac{1}{36}$
2. B
$\frac{7}{36}$
3. C
$\frac{11}{36}$
4. D
$\frac{5}{36}$

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### Solution:

Given that two dice, one blue and one grey, are thrown at the same time.
We know that the probability is given as the ratio of the number of favorable outcomes with the total number of possible outcomes.
$P\left(E\right)=\frac{\mathit{Number of favourable outcomes n}\left(E\right)}{\mathit{Total possible outcomes n}\left(S\right)}$
There are 36 possible outcomes when two dies are thrown, the outcomes are,  $\left\{\left(1,1\right),\left(1,2\right),\left(1,3\right),\left(1,4\right),\left(1,5\right),\left(1,6\right),\left(2,1\right),\left(2,2\right),\left(2,3\right),\left(2,4\right),\left(2,5\right),\left(2,6\right),$
$\left(3,1\right),\left(3,2\right),\left(3,3\right),\left(3,4\right),\left(3,5\right),\left(3,6\right),\left(4,1\right),\left(4,2\right),\left(4,3\right)\left(4,4\right)\left(4,5\right),\left(4,6\right),$
$\left(51\right),\left(5,2\right),\left(5,3\right),\left(5,4\right),\left(5,5\right),\left(5,6\right),\left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right)\left(6,5\right)\left(6,6\right)\right\}$
Total possible outcomes, $n\left(S\right)=36$
Let E be the event that the sum of two numbers appearing on the two dice is 8.
$E=\left\{\left(2,6\right),\left(6,2\right),\left(3,5\right),\left(5,3\right),\left(4,4\right)\right\}$.
$⇒n\left(E\right)=5$
The probability of getting numbers whose sum is 8 is,
$P\left(E\right)=\frac{\mathit{Number of favourable outcomes n}\left(E\right)}{\mathit{Total number of outcomes n}\left(S\right)}$
$⇒P\left(E\right)=\frac{5}{36}$
Thus, the probability of getting numbers whose sum is 8 is $\frac{5}{36}$.
Hence, option 4 is correct.

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