Question

# $15$ cards, numbered $1,2,3,\dots ,15$ are put in a box and mixed thoroughly. A card is drawn at random from the box. Find the probability that the card drawn bears an even number, and a number divisible by $2$ or $3$.

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Solution

## Step 1: Finding the probability of drawing an even number card:The fifteen cards are numbered from $1,2,3,...15$The even numbers in the cards are $2,4,6,8,10,12,14$.Thus, the number of favorable outcomes for the given event is $7$We know that,$\text{Probability=}\frac{\text{Numberoffavorableoutcomes}}{\text{Totalnumberofoutcomes}}$Thus, the probability will be,$=\frac{7}{15}$Step 2: Finding the probability of drawing a number divisible by $2$ or $3$.:The numbers in the fifteen cards which are divisible by $2$ are $2,4,6,8,10,12,14$Thus, the number of favorable outcomes for the given event is $7$The numbers in the fifteen cards which are divisible by $3$ are $3,6,9,12,15$Thus, the number of favorable outcomes for the given event is $5$The number of outcomes that are common to both the events is $\left\{6,12\right\}$Thus, the number of favorable outcomes for the entire event is $7+5-2=10$ We know that,$\text{Probability=}\frac{\text{Numberoffavorableoutcomes}}{\text{Totalnumberofoutcomes}}$Thus, the probability will be,$=\frac{10}{15}\phantom{\rule{0ex}{0ex}}=\frac{2}{3}$Hence, the probability that the card drawn bears an even number is $\frac{7}{15}$, and a number divisible by $2$ or $3$ is $\frac{2}{3}$.

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