Question

# Two dice are thrown once. Find the probability of getting an even number on the $1st$ die or a total of $8$.

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Solution

## Step 1: Find the number of favourable outcomes:Sample space for throwing two dice will be,$n\left(S\right)={\text{(numberofpossibleoutcomesofthrowingadice)}}^{\text{numberofdicesthrown}}\phantom{\rule{0ex}{0ex}}n\left(S\right)={\left(6\right)}^{2}\phantom{\rule{0ex}{0ex}}n\left(S\right)=36$Let A be the event of getting an even number on the first die and B be the event of getting a total of $8$.The favourable outcomes of A will be,$=\left(2,1\right),\left(2,2\right),\left(2,3\right),\left(2,4\right),\left(2,5\right),\left(2,6\right),\left(4,1\right),\left(4,2\right),\left(4,3\right),\left(4,4\right),\left(4,5\right),\left(4,6\right),\left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right),\left(6,6\right)$Therefore, the number of favourable outcomes will be $18$.The favourable outcomes of B will be,$=\left(2,6\right),\left(3,5\right),\left(4,4\right),\left(5,3\right),\left(6,2\right)$Therefore, the number of favourable outcomes will be $5$.Step 2: Find the number of favourable outcomes which are common to both A and B:$A\cap B=\left(2,6\right),\left(4,4\right),\left(6,2\right)$Therefore, the number of common favourable outcomes are $n\left(A\cap B\right)=3$.Step 3: Use the formula for probability:$\text{Probability=}\frac{\text{Numberoffavorableoutcomes}}{\text{Totalnumberofoutcomes}}$The probability of each outcome will be,$P\left(A\right)=\frac{18}{36}\phantom{\rule{0ex}{0ex}}P\left(B\right)=\frac{5}{36}\phantom{\rule{0ex}{0ex}}P\left(A\cap B\right)=\frac{3}{36}$Thus, the probability will be,$P\left(AUB\right)=P\left(A\right)+P\left(B\right)–P\left(A\cap B\right)\phantom{\rule{0ex}{0ex}}=\frac{18}{36}+\frac{5}{36}-\frac{3}{36}\phantom{\rule{0ex}{0ex}}=\frac{18+5-3}{36}\phantom{\rule{0ex}{0ex}}=\frac{20}{36}\phantom{\rule{0ex}{0ex}}=\frac{5}{9}$Final answer:Hence, the probability of getting an even number on the $1st$ die or a total of $8$ is $\frac{5}{9}$.

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