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 $3$.

Answer 1

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,

$\text{Probability =}\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

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 $\{6,12\}$

Thus, the number of favorable outcomes for the entire event is $7+5-2=10$

We know that,

$\text{Probability =}\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

Thus, the probability will be,

$$\begin{gathered} =\frac{10}{15} \\ =\frac{2}{3} \end{gathered}$$

Hence, the probability that the card drawn bears an even number is $\frac{7}{15}$, and a number divisible by $2$ or $3$ is $\frac{2}{3}$.