Question

Two dice are thrown together. Find the probability of getting a sum less than $12$

Answer 1

The possible outcomes when two dice are thrown together are:

$(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)$.

Therefore, the number of possible outcomes when two dice are thrown is $36$.

Now, the possible outcomes of getting a sum less than $12$ are:

$(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5)$

which means the number of favourable outcome is $35$.

Therefore, probability $P$ of getting a sum less than $12$ is:

$P=\frac{35}{36}$

Hence, the probability of getting a sum less than $12$ is $\frac{35}{36}$.