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Expert-verified Found in: Page 97 ### A First Course in Probability

Book edition 9th
Author(s) Sheldon M. Ross
Pages 432 pages
ISBN 9780321794772 # If two fair dice are rolled, what is the conditional probability that the first one lands on 6 given that the sum of the dice is $i$? Compute for all values of $i$ between $2$ and $12$

$\begin{array}{r}P\left(A\mid i=2\right)=0\\ P\left(A\mid i=3\right)=0\\ P\left(A\mid i=4\right)=0\\ P\left(A\mid i=5\right)=0\\ P\left(A\mid i=6\right)=0\\ P\left(A\mid i=7\right)=\frac{1}{6}\\ P\left(A\mid i=8\right)=\frac{1}{5}\\ P\left(A\mid i=9\right)=\frac{1}{4}\\ P\left(A\mid i=10\right)=\frac{1}{3}\\ P\left(A\mid i=11\right)=\frac{1}{2}\\ P\left(A\mid i=12\right)=1\end{array}$

See the step by step solution

## Step 1 : Given Information

The number on first dice= $6$

Sum of both dice= $i$.

## Step 2 Calculation.

When i = $2,3,4,5,6$

Consider that $A$ is the event that first lands in $6$.

$\begin{array}{r}P\left(A\mid i=2\right)=0\\ P\left(A\mid i=3\right)=0\\ P\left(A\mid i=4\right)=0\\ P\left(A\mid i=5\right)=0\\ P\left(A\mid i=6\right)=0.\end{array}$

$\begin{array}{r}P\left(A\mid i=7\right)=\frac{1}{6}\\ \end{array}$

$\begin{array}{r}\\ P\left(A\mid i=8\right)\text{, that is}i=8\text{occurs when}\left(2,6\right)\left(6,2\right)\left(5,3\right)\left(3,5\right)\left(4,4\right)\end{array}$

Thus,

$P\left(A\mid i=9\right)\text{, that is}i=9\text{occurs when}\left(3,6\right)\left(6,3\right)\left(5,4\right)\left(4,5\right)$

$\begin{array}{r}P\left(A\mid i=8\right)=\frac{1}{5}\\ P\left(A\mid i=9\right),\text{that}\\ \text{Thus,}\\ P\left(A\mid i=9\right)=\frac{1}{4}\end{array}$

$P\left(A\mid i=10\right)$ that is $i=10$ occurs when $\left(4,6\right)\left(6,4\right)\left(5,5\right)$

Thus,

$P\left(A\mid i=10\right)=\frac{1}{3}$

$P\left(A\mid i=11\right)$ that is $i=11$ occurs when $\left(5,6\right)\left(6,5\right)$

$P\left(A\mid i=11\right)=\frac{1}{2}$

$P\left(A\mid i=12\right)$, that is $i=12$occurs when $\left(6,6\right)$

Thus,

$P\left(A\mid i=12\right)=1$ ### Want to see more solutions like these? 