Question

Each of the numbers from $1$ to $30$ is written on a card and placed in a bag. If one card is drawn at random, what is the probability that the number is a multiple of $2$ or a multiple of $3$?

Answer 1

Step- 1: The probability of a multiple of $2$:

The formula for the probability is,

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

The multiple of $2$ are $2$, $4$, $6$, $8$, $10$, $12$, $14$, $16$, $18$, $20$, $22$, $24$, $26$, $28$, and $30$.

The number of multiple of $2$ is $15$.

The number of total outcomes is $30$.

Substitute the values in the probability formula,

$\text{P}\left(\text{Multiple of}2\right)=\frac{15}{30}$

Step- 2: The probability of a multiple of $3$:

The multiple of $3$ are $3$, $6$, $9$, $12$, $15$, $18$, $21$, $24$, $27$, and $30$.

The number of multiple of $3$ is $10$.

Substitute the values in the probability formula,

$\text{P}\left(\text{Multiple of 3}\right)=\frac{10}{30}$

Step-3: The probability of multiple of $2$ and $3$:

The multiple of $2$ and $3$ are $6$, $12$, $18$, $24$, and $30$.

The number of multiple of $2$ and $3$ is $5$.

Substitute the values in the probability formula,

$\text{P}\left(\text{Multiple of 2 and 3}\right)=\frac{5}{30}$

Step- 4: The probability of a multiple of $2$ or $3$:

$\text{P}\left(\text{Multiple of 2 or 3}\right)=\text{P}\left(\text{Multiple of}2\right)+\text{P}\left(\text{Multiple of}3\right)-\text{P}\left(\text{Multiple of}2and3\right)$

Substitute the values,

$$\begin{gathered} \text{P}\left(\text{Multiple of 2 or 3}\right)=\frac{15}{30}+\frac{10}{30}-\frac{5}{30} \\ \text{P}\left(\text{Multiple of 2 or 3}\right)=\frac{20}{30} \\ \text{P}\left(\text{Multiple of 2 or 3}\right)=\frac{2}{3} \end{gathered}$$

Hence, the probability of a multiple of $2$ or $3$ is $\frac{\mathbf{2}}{\mathbf{3}}$.