Search any question & find its solution
Question:
Answered & Verified by Expert
Three dice are thrown at the same time. Find the probability of getting three two's, if it is known that the sum of the numbers on the dice was six.
Solution:
2816 Upvotes
Verified Answer
On a throw of three dice, $[\mathrm{n}(\mathrm{S})]=6^3=216$
Let $\mathrm{E}_1$ is the event when the sum of numbers on the dice was six and $E_2$ is the event when three two's occurs.
$$
\begin{aligned}
&\Rightarrow \mathrm{E}_1=\{(1,1,4),(1,2,3,),(1,3,2),(1,4,1),(2,1,3), \\
&(2,2,2),(2,3,1),(3,1,2),(3,2,1),(4,1,1,)\} \\
&\Rightarrow \mathrm{n}\left(\mathrm{E}_1\right)=10 \text { and } \mathrm{E}_2=\{2,2,2\} \\
&\Rightarrow \mathrm{n}\left(\mathrm{E}_2\right)=1 \\
&\text { Also, }\left(\mathrm{E}_1 \cap \mathrm{E}_2\right)=1 \\
&\therefore \quad \mathrm{P}\left(\mathrm{E}_2 / \mathrm{E}_1\right)=\frac{\mathrm{P} \cdot\left(\mathrm{E}_1 \cap \mathrm{E}_2\right)}{\mathrm{P}\left(\mathrm{E}_1\right)}=\frac{1 / 216}{10 / 216}=\frac{1}{10}
\end{aligned}
$$
Let $\mathrm{E}_1$ is the event when the sum of numbers on the dice was six and $E_2$ is the event when three two's occurs.
$$
\begin{aligned}
&\Rightarrow \mathrm{E}_1=\{(1,1,4),(1,2,3,),(1,3,2),(1,4,1),(2,1,3), \\
&(2,2,2),(2,3,1),(3,1,2),(3,2,1),(4,1,1,)\} \\
&\Rightarrow \mathrm{n}\left(\mathrm{E}_1\right)=10 \text { and } \mathrm{E}_2=\{2,2,2\} \\
&\Rightarrow \mathrm{n}\left(\mathrm{E}_2\right)=1 \\
&\text { Also, }\left(\mathrm{E}_1 \cap \mathrm{E}_2\right)=1 \\
&\therefore \quad \mathrm{P}\left(\mathrm{E}_2 / \mathrm{E}_1\right)=\frac{\mathrm{P} \cdot\left(\mathrm{E}_1 \cap \mathrm{E}_2\right)}{\mathrm{P}\left(\mathrm{E}_1\right)}=\frac{1 / 216}{10 / 216}=\frac{1}{10}
\end{aligned}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.