Probability: Question 1

Worked solution

Step 1: Find the probability of each event separately

Let $A$ be “the number is a multiple of $5$” and $B$ be “the number is a multiple of $8$”.

Multiples of $5$ from $1$ to $50$: $5, 10, 15, \ldots, 50$. That is $10$ numbers, so $P(A) = \frac{10}{50} = 0.2$

Multiples of $8$ from $1$ to $50$: $8, 16, 24, 32, 40, 48$. That is $6$ numbers, so $P(B) = \frac{6}{50} = 0.12$

Step 2: Check whether the events overlap

$A$ and $B$ are not mutually exclusive, because a number can be a multiple of both $5$ and $8$ at once (a multiple of $\text{lcm}(5,8) = 40$). In the range $1$ to $50$, the only such number is $40$ itself, so $P(A \cap B) = \frac{1}{50} = 0.02$

Step 3: Apply the addition rule

Since the events are not mutually exclusive, the overlap must be subtracted once so it is not double-counted: $P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.2 + 0.12 - 0.02 = 0.3$

Step 4: Check by listing the numbers directly

Multiples of $5$: $5, 10, 15, 20, 25, 30, 35, 40, 45, 50$. Multiples of $8$: $8, 16, 24, 32, 40, 48$.

Combining these lists and removing the repeated $40$ gives the $15$ distinct numbers $5, 8, 10, 15, 16, 20, 24, 25, 30, 32, 35, 40, 45, 48, 50$ so $P(A \cup B) = \frac{15}{50} = 0.3$ which agrees exactly with Step 3.

Why the other options are wrong

• B ($0.32$): this is $0.2 + 0.12$ with no subtraction for the overlap. It treats the events as mutually exclusive when they are not.
• C ($0.20$): this is only $P(A)$, the probability of a multiple of $5$, ignoring $B$ entirely.
• D ($0.12$): this is only $P(B)$, the probability of a multiple of $8$, ignoring $A$ entirely.

Final answer

• $P(\text{multiple of }5\text{ or }8) = \boxed{0.3}$, option A.