Question

Two fair dice are rolled simultaneously. The probability that $5$ will come up at least once is:

Answer 1

Probability of getting $5$ at least once when two dice are rolled simultaneuously.

The probability of any event $E$ is the ratio of the number of favorable outcomes to the total number of sample spaces.

i.e., $P\left(E\right)=\frac{\mathrm{Number}\mathrm{of}\mathrm{favourable}\mathrm{outcomes}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{outcomes}}$

The number of sides in a die is $6$.

The total outcome for rolling two fair dice simultaneously is $6\times 6=36$

The possible outcomes for getting at least $5$ by rolling a fair dice twice are:

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

So, the number of favorable outcomes is $11$

Consider $E$ to be the probability of getting $5$ at least once.

Probability of getting $5$ at least once:

$$\begin{gathered} P\left(E\right)=\frac{\mathrm{Number}\mathrm{of}\mathrm{favourable}\mathrm{outcomes}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{outcomes}} \\ =\frac{11}{36} \end{gathered}$$

Hence, the probability that 5 will come up at least once is $\frac{11}{36}$.