Question

When two dice are rolled, find the probability of getting a greater number on the first die than the one on the second, given that the sum should equal 8.

• A. $\frac{1}{5}$
• B. $\frac{2}{5}$
• C. $\frac{3}{5}$
• D. None of these

Answer 1

Answer: (B)

$\frac{2}{5}$

Let the event of getting a greater number on the first die be $G$.

There are $5$ ways to get a sum of $8$ when two dice are rolled $=\left\{\right(2,6),(3,5),(4,4),(5,3),(6,2\left)\right\}$

And there are two ways where the number on the first die is greater than the one on the second given that the sum should equal $8$, $G=\left\{\right(5,3),(6,2\left)\right\}$

Therefore, $P($Sum equals $8)$ $=\frac{5}{36}$ and $P\left(G\right)$ $=\frac{2}{36}$

Now, $P(G|$ Sum equals $8)$ $=\frac{P\left(Gandsumequals8\right)}{P\left(sumequals8\right)}$

$=\frac{\left(2/36\right)}{\left(5/36\right)}$ $=\frac{2}{5}$