If A and B are any two events such that P(B) = P(A and B), then which of the following is correct Option 1: P(B|A) = 1 Option 2: P(A|B) = 1 Option 3: P(B|A) = 0 Option 4: P(A|B) = 0

Answer: P(A|B) = 1. Solution: We are given that P(B) = P(A B) and need to determine the relationship between events A and B. P(A B) is the probability that both A and B happen simultaneously. Normally, P(A B) ≤ P(B), because "both A and B happening" is either equal to or a smaller portion than "B happening." But here they are equal — meaning every time B occurs, A also occurs. B cannot happen without A happening. Visually, the circle of B sits entirely inside the circle of A: B A Using the conditional probability formula: P(A, B) = (P(A B))/(P(B)) Since P(A B) = P(B), we substitute: P(A, B) = (P(B))/(P(B)) = 1 Also: P(B, A) = (P(A B))/(P(A)) = (P(B))/(P(A)) Since B A, we know P(B) ≤ P(A), which gives us: P(B, A) ≤ 1 Therefore: P(B, A) ≤ P(A, B) The answer is P(A, B) = 1, which equivalently means B A. Note: This does not mean P(B, A) = 1, because A can occur without B occurring — A is the larger set. Knowing A happened does not guarantee B happened. Also, this does not imply A and B are independent. Independence requires P(A B) = P(A) * P(B), which is a different condition entirely.

Algebra

Probability

Medium

core

If A and B are any two events such that P(B) = P(A and B), then which of the following is correct

$P(B|A) = 1$

$P(A|B) = 1$

$P(B|A) = 0$

$P(A|B) = 0$

✅ Correct Option: 2

We are given that $P(B) = P(A \cap B)$ and need to determine the relationship between events $A$ and $B$ .

$P(A \cap B)$ is the probability that both $A$ and $B$ happen simultaneously.

Normally, $P(A \cap B) \leq P(B)$ , because "both $A$ and $B$ happening" is either equal to or a smaller portion than " $B$ happening."

But here they are equal — meaning every time $B$ occurs, $A$ also occurs. $B$ cannot happen without $A$ happening.

Visually, the circle of $B$ sits entirely inside the circle of $A$ :

$B \subseteq A$

Using the conditional probability formula:

$P(A | B) = \dfrac{P(A \cap B)}{P(B)}$

Since $P(A \cap B) = P(B)$ , we substitute:

$P(A | B) = \dfrac{P(B)}{P(B)} = 1$

Also:

$P(B | A) = \dfrac{P(A \cap B)}{P(A)} = \dfrac{P(B)}{P(A)}$

Since $B \subseteq A$ , we know $P(B) \leq P(A)$ , which gives us:

$P(B | A) \leq 1$

Therefore:

$P(B | A) \leq P(A | B)$

The answer is $P(A|B) = 1$ , which equivalently means $B \subseteq A$ .

Note: This does not mean $P(B|A) = 1$ , because $A$ can occur without $B$ occurring — $A$ is the larger set. Knowing $A$ happened does not guarantee $B$ happened.

Also, this does not imply $A$ and $B$ are independent. Independence requires $P(A \cap B) = P(A) \cdot P(B)$ , which is a different condition entirely.

2025: 14 May Shift 1

If A is any event associated with sample space and If $E_1, E_2, E_3$ are mutually exclusive and exhaustive events. Then which of the following are true?

(A) $P(A) = P(E_1)P(E_2|A) + P(E_2)P(E_2|A) + P(E_3)P(E_3|A)$

(B) $P(A) = P(A|E_1)P(E_1) + P(A|E_2)P(E_2) + P(A|E_3)P(E_3)$

(D) $P(A|E_i) = \frac{P(E_i|A)P(E_i)}{\sum_{i=1}^3 P(E_i|A)P(E_i)}$ , $i = 1,2,3$

Choose the correct answer from the options given below:

2026: 29 May Shift 2

An insurance company insured 200 scooter drivers, 400 car drivers and 600 Truck drivers. The probability of an accident involving scooter driver, car driver and truck driver are 0.01, 0.03 and 0.15 respectively. One of the insured person meets an accident, then the probability that he is a truck driver is:-

2025: 26 May Shift 2

Let A, B, C be three events. If the probability of occurring exactly one out of A and B is $\frac{3}{5}$ , exactly one of B and C is $\frac{1}{5}$ , exactly one of C and A is $\frac{3}{5}$ and that of occurring of three events is $\frac{4}{25}$ , then the probability of occurring at least one of them is

2025: 21 May Shift 2

The probability distribution function of a normal variate with mean $\mu$ and variance $\sigma^2$ is given by:

$f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$ , $-\infty < x < \infty$ , $-\infty < \mu < \infty$ , $\sigma > 0$

If $y = f(x)$ be the normal probability curve, then which of the following is correct?

(A) The normal curve is symmetrical about the line $x = \mu$ .

(B) Mean, median and mode of the distribution coincide.

(D) If x increases numerically, $f(x)$ decreases rapidly.

Choose the correct answer from the options given below:

2024: 16 May Shift 1

A die is rolled thrice. What is the probability of getting a number greater than $4$ in the first and the second throw of dice and a number less than $4$ in the third throw ?

2022: 17 Aug Shift 2

In a hospital, there are 300 patients, out of which 120 are female. It is known that out of 120 females, 10% of the patients are below 40 years of age. What is the probability that a patient chosen randomly is below 40 yrs of age given that the chosen patient is a female.

If A and B are any two events such that P(B) = P(A and B), then which of the following is correct

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