Two integers are selected at random from the set {1, 2,...., 11}. Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is :
- $\frac{2}{5}$
- $\frac{1}{2}$
- $\frac{3}{5}$
- $\frac{7}{10}$
The Correct Option is A
Solution and Explanation
Top Questions on Conditional Probability
Given three identical bags each containing 10 balls, whose colours are as follows:
Bag I 3 Red 2 Blue 5 Green Bag II 4 Red 3 Blue 3 Green Bag III 5 Red 1 Blue 4 Green A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from Bag I is $ p $ and if the ball is Green, the probability that it is from Bag III is $ q $, then the value of $ \frac{1}{p} + \frac{1}{q} $ is:
- JEE Main - 2025
- Mathematics
- Conditional Probability
- A bag contains 19 unbiased coins and one coin with heads on both sides. One coin is drawn at random and tossed, and heads turns up. If the probability that the drawn coin was unbiased is $ \frac{m}{n} $, where $ \gcd(m, n) = 1 $, then $ n^2 - m^2 $ is equal to:
- JEE Main - 2025
- Mathematics
- Conditional Probability
- Probability of event A is 0.7 and event B is 0.4, and $ P(A \cap B^c) = 0.5 $, then the value of $ P(B | A \cup B^c) $ is equal to:
- JEE Main - 2025
- Mathematics
- Conditional Probability
- \(\text{Let } X \text{ denote the number of hours you play during a randomly selected day. The probability that } X \text{ can take values } x \text{ has the following form, where } c \text{ is some constant:}\)
\(P(X = x) = \begin{cases} 0.1, & \text{if } x = 0 \\ cx, & \text{if } x = 1 \text{ or } x = 2 \\ c(5 - x), & \text{if } x = 3 \text{ or } x = 4 \\ 0, & \text{otherwise} \end{cases}\)
\(\text{Match List-I with List-II:}\)
- CUET (UG) - 2024
- Mathematics
- Conditional Probability
- A, B, C are mutually exclusive and exhaustive events of a random experiment and E is an event that occurs in conjunction with one of the events A, B, C. The conditional probabilities of E given the happening of A, B, C are respectively 0.6, 0.3 and 0.1. If \( P(A) = 0.30 \) and \( P(B) = 0.50 \), then \( P(C \mid E) = \):
- AP EAPCET - 2024
- Mathematics
- Conditional Probability
Questions Asked in JEE Main exam
- A force of 49 N acts tangentially at the highest point of a sphere (solid of mass 20 kg) kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is:
- JEE Main - 2025
- Quantum Mechanics
Let \( A = \{-3, -2, -1, 0, 1, 2, 3\} \). A relation \( R \) is defined such that \( xRy \) if \( y = \max(x, 1) \). The number of elements required to make it reflexive is \( l \), the number of elements required to make it symmetric is \( m \), and the number of elements in the relation \( R \) is \( n \). Then the value of \( l + m + n \) is equal to:
- JEE Main - 2025
- Sets and Relations
For hydrogen-like species, which of the following graphs provides the most appropriate representation of \( E \) vs \( Z \) plot for a constant \( n \)?
[E : Energy of the stationary state, Z : atomic number, n = principal quantum number]- JEE Main - 2025
- Hydrogen
- Concentrated nitric acid is labelled as 75%by mass. The volume in mL of the solution which contains 30 g of nitric acid is: Given: Density of nitric acid solution is 1.25 g/mL.
- JEE Main - 2025
- Solutions
The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is $ 4 \_\_\_\_\_$.
- JEE Main - 2025
- permutations and combinations
Concepts Used:
Conditional Probability
Conditional Probability is defined as the occurrence of any event which determines the probability of happening of the other events. Let us imagine a situation, a company allows two days’ holidays in a week apart from Sunday. If Saturday is considered as a holiday, then what would be the probability of Tuesday being considered a holiday as well? To find this out, we use the term Conditional Probability.
Let’s discuss certain theorems of Conditional Probability:
- Let us consider a random experiment where the sample space S is considered as space and two events namely A and B happen there. Then, the formula would be:
P(S | B) = P(B | B) = 1.
Proof of the same: P(S | B) = P(S ∩ B) ⁄ P(B) = P(B) ⁄ P(B) = 1.
[S ∩ B indicates the outcomes common in S and B equals the outcomes in B].
- Now let us consider any two events namely A and B happening in a sample space ‘s’, then, P(A ∩ B) = P(A).
P(B | A), P(A) >0 or, P(A ∩ B) = P(B).P(A | B), P(B) > 0.
This theorem is named as the Multiplication Theorem of Probability.
Proof of the same: As we all know that P(B | A) = P(B ∩ A) / P(A), P(A) ≠ 0.
We can also say that P(B|A) = P(A ∩ B) ⁄ P(A) (as A ∩ B = B ∩ A).
So, P(A ∩ B) = P(A). P(B | A).
Similarly, P(A ∩ B) = P(B). P(A | B).
The interesting information regarding the Multiplication Theorem is that it can further be extended to more than two events and not just limited to the two events. So, one can also use this theorem to find out the conditional probability in terms of A, B, or C.
Read More: Types of Sets
Sometimes students get confused between Conditional Probability and Joint Probability. It is essential to know the differences between the two.