Question

A box contains $30$ cards, numbered from $1$ to $30$. A card is drawn from the box at random. Find the probability that the number on the drawn card is even, prime, a multiple of $7$.

Answer 1

Step 1: Write the formula for probability and find the total number of outcomes:

The formula for probability is,

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

Given that, the box contains $30$ cards, numbered from $1$ to $30$.

Thus, the total number of outcomes will be $30.$

Step 2: Find the probability of the number drawn on the cards is an even number:

Favorable outcomes: $2,4,6,8,10,12,14,16,18,20,22,24,26,28,30$

Number of favorable outcome$=15$

Total number of outcomes $=30$

Therefore, probability of the number drawn on the cards is an even number

$$\begin{gathered} P(evennumber)=\frac{15}{30} \\ =\frac{1}{2} \end{gathered}$$

Step 3: Find the probability of the number drawn on the cards is a prime number:

Favorable outcomes: $2,3,5,7,11,13,17,19,23,29$

Number of favorable outcome$=10$

Total number of outcomes $=30$

Therefore, probability of the number drawn on the cards is a prime number

$$\begin{gathered} P(Primenumber)=\frac{10}{30} \\ =\frac{1}{3} \end{gathered}$$

Step 4: Find the probability of the number drawn on the cards is a multiple of 7:

Favorable outcomes:$7,14,21,28$

Number of favorable outcome$=4$

Total number of outcomes $=30$

Therefore, probability of the number drawn on the cards is a multiple of $7$

$$\begin{gathered} P(Multipleof7)=\frac{4}{30} \\ =\frac{2}{15} \end{gathered}$$

Final answer:

Probability that the number on the drawn card is

(i) Even=$\frac{1}{2}$

(ii) Prime Number=$\frac{1}{3}$

(iii) multiple of $7$ $=\frac{2}{15}$