Consider the following statements for three events A, B and C: I. \(P(A\cap B\cap C) \le P(A)+P(B)+P(C)-2\) II. \(P(A\cup B\cup C) \ge P(A)+P(B)+P(C)\) Which of the statements given above is/are correct?
Neither I nor II
By subadditivity, \(P(A\cup B\cup C) \le P(A)+P(B)+P(C)\) always holds, so statement II (with \(\ge\)) is false in general — e.g. taking \(A=B=C\) with probability 0.5 gives \(0.5 \ge 1.5\), which is false. By the Bonferroni inequality, \(P(A\cap B\cap C) \ge P(A)+P(B)+P(C)-2\) always holds, so statement I (with \(\le\)) is also false in general — e.g. \(P(A)=P(B)=P(C)=0.9\) for the same event gives \(0.9 \le 0.7\), which is false. Hence neither statement is correct.
Similar Questions
Two men hit at a target with probabilities 1/2 and 1/3 respectively. What is the probability that exactly one of them hits the target?
Two similar boxes B i(i = 1, 2) contain (i + 1) red and (5 – i – 1) black balls. One box is chosen at random and two balls are drawn randomly. What is the probability that both the balls are of different colours?
Consider the following relations for two events E and F:
1. P(E ∩ F) ≥ P(E) + P(F) - 1
2. P(E ∪ F) = P(E) + P(F) + P(E ∩ F)
3. P(E ∪ F) ≤ P(E) + P(F)
Which of the above relations is/are correct?
What is \(\dfrac{P(A)+2P(B)}{3P(C)+P(D)}\) equal to?
If \(P(A\cap B) = 1/2\) and \(P(\overline{A}\cap\overline{B}) = 1/2\), and \(2P(A) = P(B) = k\), then what is the value of \(k\)?
Important Questions from Addition Theorem of Events
Two men hit at a target with probabilities 1/2 and 1/3 respectively. What is the probability that exactly one of them hits the target?
Two similar boxes B i(i = 1, 2) contain (i + 1) red and (5 – i – 1) black balls. One box is chosen at random and two balls are drawn randomly. What is the probability that both the balls are of different colours?
Consider the following relations for two events E and F:
1. P(E ∩ F) ≥ P(E) + P(F) - 1
2. P(E ∪ F) = P(E) + P(F) + P(E ∩ F)
3. P(E ∪ F) ≤ P(E) + P(F)
Which of the above relations is/are correct?
What is \(\dfrac{P(A)+2P(B)}{3P(C)+P(D)}\) equal to?
If \(P(A\cap B) = 1/2\) and \(P(\overline{A}\cap\overline{B}) = 1/2\), and \(2P(A) = P(B) = k\), then what is the value of \(k\)?
