Question

Let A and B be two independent events. The probability that both A and B occur is 1/30  and the probability that neither A nor B occurs is 2/3 The respective probabilities of A and B are

Answer 1

cannot be solved 
Total - Neither = P(a) + P(b) - P(a∩b)

Answer 2

The probability of A is 1/0.9 and B is 0.001

GIVEN

A and B are two independent event.

Probability of A and B occuring = 1/30

Probability of A and B not occuring = 2/3

TO FIND

Respective Probabilities of A and B

SOLUTION

We can simply solve the above problem as follows;

Let the Probability of A = P(A) = x

Probability of B = P(B) = y

x.y = 1/30 (equation 1)

It is given,

Probability of both the event occurring simultaneously = 1/30

P(A∩B) = P(A).P(B) = 1/30

And,

P(A' ∩ B' ) = P(A∩B)' = 1- P(AUB)

We know that,

P(AUB) = P(A) + P(B) - P(A∩B)

1/3 = P(A) + P(B) - 1/30

P(A) + P(B) = 1/3 + 1/30

P(A) + P(B) = 33/30 (equation 2)

x + y = 33/30

x = 33/30 - y

Putting the value of x in equation 1

33/30 -y . y = 1/30

(33-30y/30) y = 1/30

33y-30y² = 1/30 × 30

33y - 30y² = 1

33y - 30y² - 1 = 0

30y² - 33y + 1 = 0

y = 0.03

Putting the value of y in equation (1)

x × 0.03 = 1/30

= 0.001

Hence, The probability of A is 1/0.9 and B is 0.001

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