# Suppose you are playing a game in which two fair dice are to rolled. To make the first move, you need to roll doubles or a sum of 3 or 11. What is the probability that you will be able make the first move?

Sep 1, 2016

$\frac{5}{18} = 0.278$

#### Explanation:

so we need the probability of

$a$= rolling doubles or $b$=sum of 3 or 11.

$p \left(a \vee b\right) = p \left(a\right) + p \left(b\right)$ if they are independent.

In fact rolling a double will never be a sum of 3 or 11 so these are mutually exclusive events.
Hence, the total numbers you can get is $6 \times 6$ so no matter what you roll there are always $36$ possible outcomes.

$p \left(a\right) = \frac{6}{36}$, you can only roll a double for each number.

$p \left(b\right) = \frac{4}{36}$, you can get a sum of 3 by rolling either a 1,2 or 2,1 and the same logic for obtaining an 11 from 5,6 or 6,5.

$p \left(a \vee b\right) = \frac{6}{36} + \frac{4}{36} = \frac{10}{36} = \frac{5}{18} = 0.278$