Question

If three dice are thrown together, then the probability of getting $5$ on at least one of them is

• A. $\frac{125}{216}$
• B. $\frac{215}{216}$
• C. $\frac{1}{216}$
• D. $\frac{91}{216}$

Answer 1

The correct option is D

$\frac{91}{216}$

Explanation for the correct option:

Finding the probability:

When three dice are thrown together, the total number of outcomes $\begin{array}{rcl}& =& 6^3\\ & =& 216\end{array}$

The probability of not getting $5$ on a single die $=\frac{5}{6}$

The probability of not getting $5$ in all three dice $\begin{array}{rcl}& =& {\left(\frac{5}{6}\right)}^3=\frac{125}{216}\end{array}$

Therefore,

$\begin{array}{rcl}\mathrm{P}\left(\mathrm{getting}5\mathrm{on}\mathrm{atleast}\mathrm{one}\mathrm{die}\right)& =& 1-\mathrm{the}\mathrm{probability}\mathrm{of}\mathrm{not}\mathrm{getting}5\mathrm{in}\mathrm{all}\mathrm{three}\mathrm{dice}\\ & =& 1-\frac{125}{216}\\ & =& \frac{91}{216}\end{array}$

Hence, the correct option is D.