If \(P(A)=\frac 35\) and \(P(B)=\frac 15\), find \(P(A∩B\)) if A and B are independent events.
If \(P(A)=\frac 35\) and \(P(B)=\frac 15\), find \(P(A∩B\)) if A and B are independent events.
Solution and Explanation
As A and B are independent events. Therefore,
\(P(A∩B)=P(A).P(B)\)
\(P(A∩B) =\frac {3}{5}×\frac {1}{5}\)
\(P(A∩B) =\frac {3}{25}\)
Top Questions on Probability
- A bag contains 30 balls out of which 'm' number of balls are blue in colour.
(i) Find the probability that a ball drawn at random from the bag is not blue.
(ii) If 6 more blue balls are added in the bag, then the probability of drawing a blue ball will be \(\frac{5}{4}\) times the probability of drawing a blue ball in the first case. Find the value of m.
- CBSE Class X - 2026
- Mathematics
- Probability
- A box contains 5 red and 7 white balls. Two balls are drawn at random. What is the probability that both are red?
- A random variable \( X \) takes values \( 0, 1, 2, 3 \) with probabilities \( \frac{2a+1}{30}, \frac{8a-1}{30}, \frac{4a+1}{30}, b \) respectively, where \( a, b \in \mathbb{R} \). Let \( \mu \) and \( \sigma \) respectively be the mean and standard deviation of \( X \) such that \( \sigma^2 + \mu^2 = 2 \). Then \( \frac{a}{b} \) is equal to :
- JEE Main - 2026
- Mathematics
- Probability
- A bag contains 10 balls out of which \( k \) are red and \( (10-k) \) are black, where \( 0 \le k \le 10 \). If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is:
- JEE Main - 2026
- Mathematics
- Probability
- If \( \alpha \) and \( \beta \) (\( \alpha<\beta \)) are the roots of the equation \( (-2 + \sqrt{3})(\sqrt{x} - 3) + (x - 6\sqrt{x}) + (9 - 2\sqrt{3}) = 0 \), \( x \ge 0 \), then \( \sqrt{\frac{\beta}{\alpha}} + \sqrt{\alpha\beta} \) is equal to:
- JEE Main - 2026
- Mathematics
- Probability
Questions Asked in CBSE CLASS XII exam
- If \[ P = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix} \quad \text{and} \quad Q = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 1 & -1 & 5 \end{bmatrix} \] find \( QP \) and hence solve the following system of equations using matrix method:
\[ x - y = 3,\quad 2x + 3y + 4z = 13,\quad y + 2z = 7 \] - How will you confirm the presence of five –OH groups in a glucose molecule, which are attached to different carbon atoms?
- CBSE CLASS XII - 2026
- Biomolecules
- For a reaction : \( N_2 + 3H_2 \rightarrow 2NH_3 \), the rate of reaction with respect to \( NH_3 \) is
- CBSE CLASS XII - 2026
- Chemical Kinetics
- Arora and Gurmeet were partners in a firm sharing profits and losses in the ratio of 3 : 2. Starting from 1st October, 2024 Arora withdrew ₹ 30,000 at the beginning of each quarter for his personal use. Interest on drawings was to be charged @ 12% per annum. Interest on Arora's drawings for the year ended 31st March, 2025 was:
- CBSE CLASS XII - 2026
- Partnership
The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that
(i) target is hit.
(ii) at least one shot misses the target.
- CBSE CLASS XII - 2026
- Probability
Concepts Used:
Independent Events
Independent Events are those events that are not dependent on the occurrence or happening of any other event. For instance, if we flip a dice and get 2 as the outcome, and if we flip it again and then get 6 as the outcome. In Both cases, the events have different results and are not dependent on each other.
All the events that are not dependent on the occurrence and nonoccurrence are denominated as independent events. If Event 1 does not depend on the occurrence of Event 2, then both Events 1 and 2 are independent Events.
Two Events: Event 1 and Event 2 are independent if,
P(2|1) = P (2) given P (1) ≠ 0
and
P (1|2) = P (1) given P (2) ≠ 0
Two events 1 and 2 are further independent if,
P(1 ∩ 2) = P(1) . P (2)