Chapter 6: Theory of Probability (CAIIB – Paper 1)

1. If an unbiased coin is tossed once, what is the probability of getting a head?

• A. 0
• B. 2
• C. 0.5
• D. 1

Probability = (Number of favorable outcomes ÷ Total outcomes). For a coin, favorable = 1 (Head), total = 2 (Head, Tail). So, P(H) = 1/2 = 0.5.

2. Which of the following is the correct mathematical definition of probability?

• A. P(E) = Number of favorable outcomes ÷ Total number of equally likely outcomes
• B. P(E) = Number of favorable outcomes × Total number of outcomes
• C. P(E) = Number of trials ÷ Number of favorable outcomes
• D. P(E) = Number of favorable outcomes ÷ Number of trials

The classical definition of probability states that if all outcomes are equally likely, then P(E) = (Favorable outcomes / Total outcomes).

3. Two dice are thrown together. What is the probability of getting a sum of 7?

• A. 5/36
• B. 6/36
• C. 7/36
• D. 1/6

Total outcomes = 36. Favourable outcomes for sum = 7 are (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6 outcomes. So, P = 6/36 = 1/6.

4. If A and B are two independent events, then P(A ∩ B) equals:

• A. P(A) + P(B)
• B. P(A) − P(B)
• C. P(A)/P(B)
• D. P(A) × P(B)

For independent events, probability of intersection = product of probabilities, i.e., P(A ∩ B) = P(A) × P(B).

5. If a card is drawn from a standard 52-card deck, what is the probability that it is either a King or a Heart?

• A. 16/52
• B. 17/52
• C. 13/52
• D. 4/52

Total Kings = 4, Total Hearts = 13. But King of Hearts is common, so avoid double counting. Therefore favorable = 4 + 13 − 1 = 16. Probability = 16/52.

6. The conditional probability P(A|B) is defined as:

• A. P(A) + P(B)
• B. P(A) − P(B)
• C. P(A ∩ B) / P(B)
• D. P(A) × P(B)

Conditional probability of A given B is P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.

7. Which of the following is true about a random variable?

• A. It is always a fixed number
• B. It cannot take numerical values
• C. It is unrelated to probability
• D. It assigns numerical values to outcomes of a random experiment

A random variable is a rule that assigns a real number to each outcome in a sample space. It may be discrete or continuous.

8. A fair die is rolled. Let X be the random variable representing the number on the die. What is the probability distribution of X?

• A. P(X=k) = 1/3 for k = 1,2,3,4,5,6
• B. P(X=k) = 1/6 for k = 1,2,3,4,5,6
• C. P(X=k) = k/6 for k = 1,2,3,4,5,6
• D. P(X=k) = 0 for k = 1,2,3,4,5,6

Since each outcome is equally likely on a fair die, probability of each face = 1/6. So, distribution is uniform: P(X=k)=1/6.

9. Which of the following is NOT a valid property of a probability distribution of a discrete random variable?

• A. 0 ≤ P(X=x) ≤ 1 for every x
• B. Σ P(X=x) = 1
• C. Σ P(X=x) = any value greater than 1
• D. Probabilities are assigned to each possible value of X

For a valid probability distribution: each probability lies between 0 and 1, and total probability across all values must equal 1. Hence, ΣP(X=x) > 1 is invalid.

10. A random variable X has the following probability distribution:
X: 0, 1, 2, 3
P(X): 0.1, 0.3, 0.4, 0.2
What is P(X ≥ 2)?

• A. 0.6
• B. 0.4
• C. 0.3
• D. 0.2

P(X ≥ 2) = P(X=2) + P(X=3) = 0.4 + 0.2 = 0.6.

11. In probability distributions, the expected value (mean) of a discrete random variable is calculated as:

• A. Σ (X ÷ P(X))
• B. Σ (X − P(X))
• C. Σ (X)
• D. Σ [X × P(X)]

Expected value or mean of a discrete random variable = Σ [x × P(x)] across all possible values of x.

12. A random variable X takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3 respectively. What is the expected value E(X)?

• A. 1.5
• B. 2.1
• C. 2.5
• D. 3.0

E(X) = Σ [x × P(x)] = (1×0.2) + (2×0.5) + (3×0.3) = 0.2 + 1.0 + 0.9 = 2.1.

13. The expectation (mean) of a discrete random variable X is defined as:

• A. E(X) = Σ P(X)
• B. E(X) = Σ X
• C. E(X) = Σ [x × P(x)]
• D. E(X) = Σ [P(x) ÷ x]

Expectation or mean is the weighted average of all possible values of X with their probabilities, i.e., E(X) = Σ [x × P(x)].

14. A random variable X takes values 2, 4, 6 with probabilities 0.2, 0.5, 0.3 respectively. What is E(X)?

• A. 4.4
• B. 5.0
• C. 3.8
• D. 6.0

E(X) = (2×0.2) + (4×0.5) + (6×0.3) = 0.4 + 2.0 + 1.8 = 4.2 (Correction: Typo, correct answer is 4.2 not 4.4).

15. The variance of a random variable X is defined as:

• A. Var(X) = Σ [x × P(x)]
• B. Var(X) = Σ [x² × P(x)]
• C. Var(X) = Σ [x × P(x)]²
• D. Var(X) = E(X²) − [E(X)]²

Variance measures the dispersion of a random variable: Var(X) = E(X²) − [E(X)]².

16. A random variable X has values 1, 2, 3 with probabilities 0.2, 0.5, 0.3 respectively. Find Var(X).

• A. 0.25
• B. 0.41
• C. 0.50
• D. 1.20

E(X) = (1×0.2)+(2×0.5)+(3×0.3)=0.2+1.0+0.9=2.1. E(X²)=(1²×0.2)+(2²×0.5)+(3²×0.3)=0.2+2.0+2.7=4.9. Var(X)=E(X²)−[E(X)]²=4.9−(2.1)²=4.9−4.41=0.49≈0.50.

17. Standard deviation of a random variable X is defined as:

• A. √Var(X)
• B. Var(X)²
• C. E(X²) ÷ E(X)
• D. E(X)²

Standard deviation is the positive square root of variance: σ = √Var(X).

18. A random variable has E(X) = 10 and Var(X) = 25. What is the standard deviation of X?

• A. 10
• B. 2
• C. 20
• D. 5

Standard deviation = √Var(X) = √25 = 5.

19. In a Binomial distribution, if the probability of success is p, then the probability of failure is:

• A. p²
• B. 1/p
• C. 1 − p
• D. p − 1

In a binomial distribution, each trial has two outcomes: success with probability p and failure with probability q = 1 − p.

20. A coin is tossed 3 times. What is the probability of getting exactly 2 heads? (Use Binomial distribution)

• A. 3/8
• B. 3/8
• C. 1/8
• D. 1/2

Binomial formula: P(X=k)=C(n,k) p^k q^(n−k). Here n=3, k=2, p=0.5. P(2 heads)=C(3,2)(0.5)²(0.5)¹=3×0.25×0.5=0.375=3/8.

21. For a Binomial distribution with parameters n and p, the mean and variance are respectively:

• A. Mean = n, Variance = p
• B. Mean = np², Variance = np
• C. Mean = np, Variance = p(1 − p)
• D. Mean = np, Variance = np(1 − p)

For Binomial(n, p): Mean = np, Variance = npq = np(1 − p).

22. The Poisson distribution is generally used as an approximation to Binomial distribution when:

• A. n is large and p is small
• B. n is small and p is large
• C. n is small and p is 0.5
• D. n is large and p is close to 0.5

Poisson distribution approximates Binomial when n → ∞, p → 0, such that np = λ remains finite.

23. If events follow a Poisson distribution with mean λ = 4, what is the probability of exactly 2 events?

• A. 0.1465
• B. 0.1465
• C. 0.1839
• D. 0.2381

Poisson formula: P(X=k) = e^(−λ) λ^k / k! With λ=4, k=2: P= e^(−4)×4²/2!= e^(−4)×16/2=8e^(−4) ≈ 0.1465.

24. In a Poisson distribution with mean λ, both the mean and variance are equal to:

• A. Mean = λ, Variance = λ²
• B. Mean = λ², Variance = λ
• C. Mean = √λ, Variance = λ
• D. Mean = λ, Variance = λ

A key property of Poisson distribution is that mean = variance = λ.

25. In a normal distribution, the mean, median, and mode are:

• A. All equal
• B. Mean > Median > Mode
• C. Mean < Median < Mode
• D. Cannot be compared

In a perfectly normal distribution, the curve is symmetric. Hence, Mean = Median = Mode.

26. In a standard normal distribution, the mean and standard deviation are:

• A. Mean = 1, SD = 1
• B. Mean = 0, SD = 2
• C. Mean = 2, SD = 0
• D. Mean = 0, SD = 1

Standard normal distribution (Z-distribution) has mean = 0 and standard deviation = 1.

27. In a normal distribution, approximately what percentage of data lies within ±2 standard deviations from the mean?

• A. 68%
• B. 95%
• C. 99.7%
• D. 75%

Empirical Rule (68–95–99.7 Rule): ±1σ → 68%, ±2σ → 95%, ±3σ → 99.7%.

28. Credit risk is primarily concerned with:

• A. Loss due to interest rate fluctuations
• B. Loss due to foreign exchange fluctuations
• C. Loss due to borrower’s default
• D. Loss due to operational failures

Credit risk is the risk of loss due to counterparty (borrower) failing to meet its obligations.

29. The formula for Expected Loss (EL) in credit risk is:

• A. EL = EAD × (1 − PD)
• B. EL = LGD × (1 − PD)
• C. EL = EAD × LGD
• D. EL = EAD × PD × LGD

Expected Loss = Exposure at Default (EAD) × Probability of Default (PD) × Loss Given Default (LGD).

30. If a loan of ₹10 crore has PD = 2%, LGD = 40%, what is the Expected Loss?

• A. ₹0.08 crore
• B. ₹0.2 crore
• C. ₹0.4 crore
• D. ₹0.8 crore

EL = EAD × PD × LGD = 10 × 0.02 × 0.40 = 0.08 crore.

31. Value at Risk (VaR) primarily measures:

• A. Expected average loss in normal times
• B. Maximum possible loss in worst-case scenario
• C. Maximum expected loss at a given confidence level
• D. Loss that has already occurred

VaR estimates the maximum potential loss in value of a portfolio at a specified confidence level (e.g., 95% or 99%) over a given time horizon.

32. If 1-day 99% VaR of a portfolio is ₹50 lakh, it means:

• A. The portfolio will lose at least ₹50 lakh every day
• B. There is a 1% chance that the portfolio could lose more than ₹50 lakh in one day
• C. The average daily loss will be ₹50 lakh
• D. 99% of days the portfolio will earn ₹50 lakh profit

1-day 99% VaR = ₹50 lakh means with 99% confidence, the loss will not exceed ₹50 lakh in one day. There is 1% probability that losses could exceed this amount.

33. Which of the following is NOT a method of calculating VaR?

• A. Variance–Covariance Method
• B. Historical Simulation
• C. Monte Carlo Simulation
• D. Black-Scholes Method

Black-Scholes is used for option pricing, not for VaR. The common VaR methods are: Variance–Covariance, Historical Simulation, and Monte Carlo Simulation.

34. The Black-Scholes model is used for:

• A. Valuation of options
• B. Pricing of bonds
• C. Measuring credit risk
• D. Forecasting exchange rates

The Black-Scholes model provides a theoretical estimate of option prices based on variables such as stock price, strike price, time to maturity, risk-free rate, and volatility.

35. In the Black-Scholes option pricing model, which factor increases the value of a call option?

• A. Decrease in volatility
• B. Decrease in time to maturity
• C. Increase in underlying asset price
• D. Increase in strike price

A call option gives the right to buy the underlying asset at the strike price. Hence, as the asset price increases relative to the strike price, the value of the call option rises.

36. Which of the following is TRUE regarding Put-Call Parity?

• A. It applies only to American options
• B. It applies only when volatility is zero
• C. It is used to calculate Value at Risk
• D. It shows the relationship between call price, put price, stock price, and present value of strike price

Put-Call Parity: C + PV(K) = P + S where C = Call option price, P = Put option price, S = Spot price, PV(K) = Present value of strike price. It ensures no-arbitrage relationship.