Question:
Let A, B and C be three events such that the probability that exactly one of A and B occurs is \((1-k)\), the probability that exactly one of B and C occurs is \((1-2k)\), the probability that exactly one of C and A occurs is \((1-k)\) and the probability of all A, B and C occur simultaneously is \(k^2\), where \(0\lt k\lt1\). Then the probability that at least one of A, B and C occur is :
Let A, B and C be three events such that the probability that exactly one of A and B occurs is \((1-k)\), the probability that exactly one of B and C occurs is \((1-2k)\), the probability that exactly one of C and A occurs is \((1-k)\) and the probability of all A, B and C occur simultaneously is \(k^2\), where \(0\lt k\lt1\). Then the probability that at least one of A, B and C occur is :
Show Hint
Add the three "exactly one" probabilities to get \(\sum P(A)-\sum P(A\cap B)\), then use inclusion-exclusion.
Updated On: Nov 4, 2026
- greater than \(\frac{1}{8}\) but less than \(\frac{1}{4}\)
- greater than \(\frac{1}{2}\)
- greater than \(\frac{1}{4}\) but less than \(\frac{1}{2}\)
- exactly equal to \(\frac{1}{2}\)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
\(P(\text{exactly one of }A,B)=P(A)+P(B)-2P(A\cap B)=1-k\), and similarly for the other pairs.
Adding: \(2\sum P(A)-2\sum P(A\cap B)=(1-k)+(1-2k)+(1-k)=3-4k\), so \(\sum P(A)-\sum P(A\cap B)=\frac{3-4k}{2}\).
\(P(A\cup B\cup C)=\sum P(A)-\sum P(A\cap B)+P(A\cap B\cap C)=\frac{3-4k}{2}+k^2=k^2-2k+\frac32=(k-1)^2+\frac12\).
For \(0\lt k\lt1\), \((k-1)^2\gt0\), so the probability is greater than \(\frac12\).
Answer: (2) greater than \(\frac{1}{2}\)
Adding: \(2\sum P(A)-2\sum P(A\cap B)=(1-k)+(1-2k)+(1-k)=3-4k\), so \(\sum P(A)-\sum P(A\cap B)=\frac{3-4k}{2}\).
\(P(A\cup B\cup C)=\sum P(A)-\sum P(A\cap B)+P(A\cap B\cap C)=\frac{3-4k}{2}+k^2=k^2-2k+\frac32=(k-1)^2+\frac12\).
For \(0\lt k\lt1\), \((k-1)^2\gt0\), so the probability is greater than \(\frac12\).
Answer: (2) greater than \(\frac{1}{2}\)
Was this answer helpful?
0
0
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
View More Questions
Top JEE Main Probability Questions
- Bag A contains 3 white, 7 red balls and bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from bag A, if the ball drawn is white, is:
- The probability that Ajay will not go to office is \(\frac{1}{5}\) and probability that Ajay and Vijay will not go to the office is \(\frac{2}{7}\), if their visits to office is independent of each other, then find the probability that Ajay will go to the office, but Vijay will not go, is
- Given set = {1, 2, 3, ..., 50} one number is selected randomly from the set. Find the probability that number is multiple of 4 or 6 or 7.
- A coin is biased so that a head is twice as likely as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is
- An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability that the first draw gives all white balls, and the second draw gives all black balls, is:
View More Questions
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
View More Questions