📐 Probability & Statistics — Practice Worksheet

Q1. A die is thrown once. The probability of getting a number greater than 4 is:

1. 1/6
2. 1/3
3. 1/2
4. 2/3

Q2. Two dice are thrown simultaneously. The probability of getting a sum of 7 is:

1. 1/6
2. 5/36
3. 7/36
4. 1/12

Q3. If P(A) = 0.4, P(B) = 0.3, and P(A ∩ B) = 0.12, then A and B are:

1. Mutually exclusive
2. Independent
3. Dependent
4. Complementary

Q4. The number of ways to arrange the letters of the word "MISSISSIPPI" is:

1. 11! / (4! 4! 2!)
2. 11!
3. 11! / (4! 4!)
4. 11! / (4! 2! 2!)

Q5. In a binomial distribution with n = 6 and p = 1/3, the mean is:

1. 1
2. 2
3. 3
4. 6

Q6. A bag contains 5 red and 3 blue balls. Two balls are drawn at random without replacement. The probability that both are red is:

1. 5/14
2. 25/64
3. 10/28
4. 5/8

Q7. If P(A) = 0.6 and P(B|A) = 0.5, then P(A ∩ B) is:

1. 0.1
2. 0.3
3. 0.5
4. 1.1

Q8. The variance of the first n natural numbers is:

1. (n² − 1)/12
2. (n + 1)/2
3. n(n + 1)/2
4. (n − 1)²/12

Q9. Three cards are drawn from a pack of 52 cards. The probability that all three are kings is:

1. 4/52 × 3/51 × 2/50
2. 1/13 × 1/13 × 1/13
3. ⁴C₃ / ⁵²C₃
4. Both A and C

Q10. The mean deviation about the mean for the data 2, 4, 6, 8, 10 is:

1. 2
2. 2.4
3. 3
4. 4

Q11. If events A and B are mutually exclusive, then P(A ∪ B) equals:

1. P(A) × P(B)
2. P(A) + P(B)
3. P(A) + P(B) − P(A ∩ B)
4. 1 − P(A) − P(B)

Q12. In a Poisson distribution with mean λ = 3, P(X = 0) is:

1. e⁻³
2. 3e⁻³
3. 1/3
4. 0

Q13. The coefficient of variation of a distribution with mean 50 and standard deviation 10 is:

1. 5%
2. 10%
3. 20%
4. 50%

Q14. A random variable X has the probability distribution P(X = k) = C(5,k)(0.4)ᵏ(0.6)⁵⁻ᵏ. The standard deviation of X is:

1. √1.2
2. 1.2
3. 2
4. √2

Q15. Bayes' theorem is used to find:

1. Prior probability
2. Posterior probability
3. Marginal probability
4. Joint probability

Q16. A box contains 10 bulbs, 3 of which are defective. If 2 bulbs are drawn at random, find the probability that (a) both are defective, (b) at least one is defective.

Q17. A coin is tossed 5 times. Find the probability of getting (a) exactly 3 heads, (b) at least 4 heads.

Q18. The marks of 8 students are: 45, 55, 60, 65, 70, 75, 80, 90. Find the mean, variance, and standard deviation.

Q19. Two factories A and B produce 60% and 40% of a product respectively. Factory A has a 2% defect rate and B has a 5% defect rate. A product is found defective. Find the probability it came from factory A (use Bayes' theorem).

Q20. In how many ways can a committee of 5 be formed from 6 men and 4 women such that at least 2 women are included?

Q21. A random variable X has the following distribution: P(X = 0) = 0.1, P(X = 1) = 0.3, P(X = 2) = 0.4, P(X = 3) = 0.2. Find E(X) and Var(X).

Q22. A student answers a multiple-choice question with 4 options. The probability that the student knows the answer is 0.6. If the student doesn't know, they guess randomly. Given that the answer is correct, find the probability that the student actually knew the answer.

Q23. Find the number of permutations of the letters of the word "ARRANGE" taken all at a time.

Q24. The probability that a student passes in Mathematics is 0.7 and in Physics is 0.8. The probability of passing in both is 0.6. Find the probability of passing in at least one subject.

Q25. In a binomial distribution, the mean is 4 and variance is 3. Find n, p, and q. Also find P(X ≥ 1).