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 \(k\)?
2/3
Since \(\overline{A}\cap\overline{B}=\overline{A\cup B}\), we get \(P(A\cup B)=1-P(\overline{A}\cap\overline{B})=1-\frac12=\frac12\). Using \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) with \(P(A)=\frac{k}{2}\) and \(P(B)=k\) gives \(\frac{k}{2}+k-\frac12=\frac12\), so \(\frac{3k}{2}=1\) and \(k=\frac{2}{3}\).
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?
Consider the following statements for three events A, B and C:
I. \(P(A\cap B\cap C) \le P(A)+P(B)+P(C)-2\)
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Which of the statements given above is/are correct?
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?
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?
