ProbabilityPYQ Sept 25Question 4181 of 187
All Questions

A number is selected at random from the first 50 natural numbers. What is the probability that it would be either a two-digit prime number or a composite number lying between 5 and 40?

Options

A0.54
B0.48
C0.64
D0.72
For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option d — 0.72

All Options:

  • A0.54
  • B0.48
  • C0.64
  • D0.72

Detailed Solution & Explanation

Let the sample space S\displaystyle S be the first 50 natural numbers, i.e., S={1,2,…,50}\displaystyle S = \{1, 2, \dots, 50\}. The total number of outcomes is N=50\displaystyle N = 50.
Let us find the numbers that satisfy either of the two given criteria:
1. **Event A: "Two-digit prime number"**:
The prime numbers between 1\displaystyle 1 and 50\displaystyle 50 are:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,472, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Of these, the two-digit prime numbers (i.e., ≥10\displaystyle \ge 10) are:
A={11,13,17,19,23,29,31,37,41,43,47}A = \{11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47\}
The number of elements in A\displaystyle A is n(A)=11\displaystyle n(A) = 11.
2. **Event B: "Composite number lying between 5 and 40"**:
"Lying between 5 and 40" means the numbers must be from 6\displaystyle 6 to 39\displaystyle 39 inclusive (a total of 39−6+1=34\displaystyle 39 - 6 + 1 = 34 numbers).
Let us identify which of these 34\displaystyle 34 numbers are prime:
Primes in this range={7,11,13,17,19,23,29,31,37}(9 primes)\text{Primes in this range} = \{7, 11, 13, 17, 19, 23, 29, 31, 37\} \quad (\text{9 primes})
All other integers in this range are composite (excluding 1, which is not in this range). Therefore, the number of composite numbers is:
n(B)=34−9=25n(B) = 34 - 9 = 25
3. **Checking for overlap (A∩B\displaystyle A \cap B)**:
A number cannot be both prime (Event A) and composite (Event B). Therefore, events A\displaystyle A and B\displaystyle B are mutually exclusive: A∩B=∅\displaystyle A \cap B = \emptyset.
4. **Number of favorable outcomes** (n(A∪B)\displaystyle n(A \cup B)):
n(A∪B)=n(A)+n(B)=11+25=36n(A \cup B) = n(A) + n(B) = 11 + 25 = 36
5. **Probability Calculation**:
P(A∪B)=n(A∪B)N=3650=0.72P(A \cup B) = \frac{n(A \cup B)}{N} = \frac{36}{50} = 0.72
Hence, **Option D** is the correct answer.

About This Chapter: Probability

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Probability Operations, Expected Value

A logic-heavy chapter dealing with random experiments, events (mutually exclusive, exhaustive), set theory probability, conditional probability, and Bayes' Theorem. It forms the basis for Theoretical Distributions.

View Official ICAI Syllabus

Exam Strategy Tip

Always draw a quick Venn Diagram or tree when faced with 'At least one' or 'Only A but not B' wording. It saves you from double-counting.

More Questions from Probability

If P(A)=12\displaystyle P(A) = \frac{1}{2}, P(B)=13\displaystyle P(B) = \frac{1}{3}, P(A∩B)=14\displaystyle P(A \cap B) = \frac{1}{4}, then the value of P(A′∪B′)\displaystyle P(A' \cup B') is

Two dice are thrown simultaneously. Find the probability that the sum of digits on the two dice would be 8 or more.

A number is selected from the first 20 natural numbers. Find the probability that it would be divisible by 3 or 7.

A father had three sons namely, Kailash, Harish and Prakash. All are above 65 years in age. Prakash happens to be the eldest while Kailash as youngest. As per the health history, it is estimated that the probability that Kailash survives another 5 years is 45\displaystyle \frac{4}{5}, Harish survives another 5 years is 35\displaystyle \frac{3}{5} and Prakash survives another 5 years is 12\displaystyle \frac{1}{2}. The probabilities that Kailash and Harish survive another 5 years is 0.46, Harish and Prakash survive another 5 years is 0.32 and Kailash and Prakash survive another 5 years is 0.48. The probability that all three sons survive another 5 years is 0.26. What shall be the probability that at least one of them survives another 5 years?

In a class, there are 15 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected is

Two cards are drawn from a pack of 52 cards. The probability that one is a spade and one is a heart; is

Ready to Master Probability?

Practice all 187 questions with instant feedback, earn XP, track your streaks, and ace your CA Foundation exam.

Start Practicing — It's Free