Question:
Let \( E \) and \( F \) be any two independent events with \( 0<P(E)<1 \) and \( 0<P(F)<1 \). Which one of the following statements is NOT TRUE?
Let \( E \) and \( F \) be any two independent events with \( 0<P(E)<1 \) and \( 0<P(F)<1 \). Which one of the following statements is NOT TRUE?
Show Hint
For independent events, the probability of both events occurring is \( P(E \cap F) = P(E) \cdot P(F) \). For complementary events, \( P(E) + P(F) = 1 \).
Updated On: Dec 12, 2025
- \( P(\text{Neither } E \text{ nor } F \text{ occurs}) = P(E^C) \cdot P(F^C) \)
- \( P(E) = 1 - P(F) \)
- \( P(E \text{ occurs but } F \text{ does not occur}) = P(E) - P(E \cap F) \)
- \( P(E \text{ occurs given that } F \text{ does not occur}) = P(E) \)
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Check each option.
Option (B) is incorrect because \( P(E) = 1 - P(F) \) only holds if \( E \) and \( F \) are complementary events, which is not given in the problem. Step 2: Conclusion.
The correct answer is (B) \( P(E) = 1 - P(F) \).
Option (B) is incorrect because \( P(E) = 1 - P(F) \) only holds if \( E \) and \( F \) are complementary events, which is not given in the problem. Step 2: Conclusion.
The correct answer is (B) \( P(E) = 1 - P(F) \).
Was this answer helpful?
0
0
Top Questions on Probability
- If P(A) = 0.12, P(B) = 0.15 and P(B/A) = 0.18, then find the value of P(A \( \cap \) B).
- UP Board XII - 2025
- Mathematics
- Probability
If A and B are two events such that \( P(A \cap B) = 0.1 \), and \( P(A|B) \) and \( P(B|A) \) are the roots of the equation \( 12x^2 - 7x + 1 = 0 \), then the value of \(\frac{P(A \cup B)}{P(A \cap B)}\)
- JEE Main - 2025
- Mathematics
- Probability
- Cards numbered 10, 11, 12, ..., 30 are kept in a box and shuffled thoroughly. Rahit draws a card at random from the box. The probability that the number on the card is a multiple of 6 or 5 is:
- If \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A|B) = \frac{2}{5} \), then find \( P(A \cup B) \).
- UP Board XII - 2025
- Mathematics
- Probability
A quadratic polynomial \( (x - \alpha)(x - \beta) \) over complex numbers is said to be square invariant if \[ (x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2). \] Suppose from the set of all square invariant quadratic polynomials we choose one at random. The probability that the roots of the chosen polynomial are equal is ___________. (rounded off to one decimal place)
- GATE CS - 2025
- Mathematics
- Probability
View More Questions
Questions Asked in IIT JAM MS exam
- A bag has 5 blue balls and 15 red balls. Three balls are drawn at random from the bag simultaneously. Then the probability that none of the chosen balls is blue equals
- IIT JAM MS - 2024
- Probability
- Suppose that the weights (in kgs) of six months old babies, monitored at a healthcare facility, have \( N(\mu, \sigma^2) \) distribution, where \( \mu \in \mathbb{R} \) and \( \sigma > 0 \) are unknown parameters. Let \( X_1, X_2, \ldots, X_9 \) be a random sample of the weights of such babies. Let \( \overline{X} = \frac{1}{9} \sum_{i=1}^{9} X_i \), \( S = \sqrt{\frac{1}{8} \sum_{i=1}^{9} (X_i - \overline{X})^2} \) and let a 95% confidence interval for \( \mu \) based on \( t \)-distribution be of the form \( (\overline{X} - h(S), \overline{X} + h(S)) \), for an appropriate function \( h \) of random variable \( S \). If the observed values of \( \overline{X} \) and \( S^2 \) are 9 and 9.5, respectively, then the width of the confidence interval is equal to __________ (round off to 2 decimal places) (You may use \( t_{9,0.025} = 2.262, t_{8,0.025} = 2.306, t_{9,0.05} = 1.833, t_{8,0.05} = 1.86 \)).
- IIT JAM MS - 2024
- Multivariate Distributions
- Let \( X_1, X_2, X_3 \) be a random sample from a Poisson distribution with mean \( \lambda \), \( \lambda > 0 \). For testing \( H_0: \lambda = \frac{1}{8} \) against \( H_1: \lambda = 1 \), a test rejects \( H_0 \) if and only if \( X_1 + X_2 + X_3 > 1 \). The power of this test is equal to __________ (round off to 2 decimal places).
- IIT JAM MS - 2024
- Standard Univariate Distributions
- Let \( X_1, X_2, \ldots, X_{10} \) be a random sample from a \( U(-\theta, \theta) \) distribution, where \( \theta \in (0, \infty) \). Let \( X_{(10)} = \max \{ X_1, X_2, \ldots, X_{10} \} \) and \( X_{(1)} = \min \{ X_1, X_2, \ldots, X_{10} \} \). If the observed values of \( X_{(10)} \) and \( X_{(1)} \) are 8 and -10, respectively, then the maximum likelihood estimate of \( \theta \) is equal to __________ (answer in integer).
- IIT JAM MS - 2024
- Estimation
- Let \( \Theta \) be a random variable having \( U(0, 2\pi) \) distribution. Let \( X = \cos \Theta \) and \( Y = \sin \Theta \). Let \( \rho \) be the correlation coefficient between \( X \) and \( Y \). Then \( 100\rho \) is equal to __________ (answer in integer).
- IIT JAM MS - 2024
- Univariate Distributions
View More Questions