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The probability of simultaneous occurrence of atleast one of two events A and B is p. If the probability that exactly one of A, B occurs is q, then P(A’) + P(B’) is equal to
  • a)
    2 – 2p + q
  • b)
    2 + 2p – q
  • c)
    3 – 3p + q
  • d)
    2 – p + q
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The probability of simultaneous occurrence of atleast one of two event...
Since, P (exactly one of A, B occurs) = q.
∴ P (A∪B)- P (A∩B) = q
⇒ p - P (A∩B)= q ⇒ P ( A∩B)= p - q
⇒ 1- P ( A'∪B') = p-q ⇒ P (A'∪B') =1- p+q
⇒ P (A') + P(B') - P (A'∩B') =1- p+ q
⇒ P (A') + P( B') =(1- p+ q) + [1- P(A ∪B)]
= (1 – p + q) + (1 – p) = 2 – 2p + q
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Community Answer
The probability of simultaneous occurrence of atleast one of two event...
To find the probability of event A occurring, we can use the formula:

P(A) = P(A and not B) + P(A and B)

Since we know the probability of exactly one of A or B occurring is q, we can rewrite this equation as:

P(A) = q + P(A and B)

We are also given that the probability of simultaneous occurrence of at least one of A and B is p. This means:

P(A or B) = p

Using the formula for the probability of the union of two events:

P(A or B) = P(A) + P(B) - P(A and B)

Substituting in the values we know:

p = P(A) + P(B) - P(A and B)

Since we are given that the probability of exactly one of A or B occurring is q, we can rewrite this equation as:

p = q + P(B) - P(A and B)

To find the probability of event A not occurring, we can subtract P(A) from 1:

P(not A) = 1 - P(A)

Since we are looking for P(A | not B), we can rewrite this equation as:

P(A | not B) = 1 - P(A)

Substituting the equation we found earlier:

P(A | not B) = 1 - (q + P(B) - P(A and B))

Simplifying:

P(A | not B) = 1 - q - P(B) + P(A and B)

Therefore, the probability of event A occurring given that event B does not occur is 1 - q - P(B) + P(A and B).
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The probability of simultaneous occurrence of atleast one of two events A and B is p. If the probability that exactly one of A, B occurs is q, then P(A’) + P(B’) is equal toa)2 – 2p + qb)2 + 2p – qc)3 – 3p + qd)2 – p + qCorrect answer is option 'A'. Can you explain this answer?
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