Probability
Probability
Allen Star Batch
Grade 12
Question:
A number is selected at random from the first twenty-five natural numbers. If it is a composite number, then it is divided by $5$. But if it is not a composite number, it is divided by $2$. The probability that there will be no remainder in the division is ______.
Step-by-Step Solution
Key Concept: Use total probability by conditioning on whether the selected number is composite or prime.
Let $C$ be the event the number is composite and $E$ be the event there is no remainder. We have $P(C) = \frac{15}{25} = \frac{3}{5}$ and $P(\bar{C}) = \frac{2}{5}$. Then $P(E|C) = \frac{4}{15}$ and $P(E|\bar{C}) = \frac{1}{10}$. By the law of total probability: $P(E) = P(E|C)P(C) + P(E|\bar{C})P(\bar{C}) = \frac{4}{15} \cdot \frac{3}{5} + \frac{1}{10} \cdot \frac{2}{5} = \frac{12}{75} + \frac{2}{50} = 0.2$.
Correct Answer: 0.20