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?
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?
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?
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?
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?
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?
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 \(k\)?
