# CONDITIONAL PROBABILITY EXAMPLES AND SOLUTIONS

## About "Conditional Probability Examples and Solutions"

Conditional Probability Examples and Solutions :

Here we are going to see some example problems on conditional probability.

Question 1 :

A problem in Mathematics is given to three students whose chances of solving it are 1/3, 1/4 and 1/5 (i) What is the probability that the problem is solved? (ii) What is the probability that exactly one of them will solve it?

Solution :

Let "A", "B" and "C" be the events of solving problems by each students respectively.

P(A)  =  1/3, P(B)  =  1/4 and P(C)  =  1/5

(i) What is the probability that the problem is solved?

P(Problem solved)  =  P(At least one solving)

=  1 - P(None solving the problem)

=  1 - P(A' n B' n C')

=  1 -  P(A') ⋅ P(B') ⋅  P(C')

 P(A')  =  1 - P(A)=  1-(1/3)P(A')  =  2/3 P(B')  =  1 - P(B)=  1-(1/4)P(B')  = 3/4 P(C')  =  1 - P(C)=  1-(1/5)P(C')  = 4/5

=  1 -  (2/3) (3/4) (4/5)

=  1 - (2/5)

=  (5 - 2) / 5

P(Problem solved)  =  3/5

(ii) What is the probability that exactly one of them will solve it

P(exactly one of them will solve it)

=  P(A' n B' n c) + P(A' n B n c') + P(A n B' n c')

=  P(A') P(B') P(C) +  P(A') P(B) P(C') +  P(A) P(B') P(C')

=  (2/3)(3/4)(1/5) + (2/3)(1/4)(4/5) + (1/3)(3/4)(4/5)

=  (6/60) + (8/60) + (12/60)

=  (6 + 8 + 12)/60

=  26/60

P(exactly one of them will solve it)  =  13/30

Question 2 :

The probability that a car being filled with petrol will also need an oil change is 0.30; the probability that it needs a new oil filter is 0.40; and the probability that both the oil and filter need changing is 0.15.

(i)  If the oil had to be changed, what is the probability that a new oil filter is needed?

(ii)  If a new oil filter is needed, what is the probability that the oil has to be changed?

Solution :

Let "A" and "B" the event of changing oil and new oil filter respectively.

P(A)  =  0.30, P(B)  =  0.40, P(AnB)  =  0.15

(i)  If the oil had to be changed, what is the probability that a new oil filter is needed?

Here we have to find the probability that a new oil filter is needed, if the oil had to be changed.

The event B depends on A.

P(B/A)  =  P(AnB)/P(A)

=  0.15 / 0.30

=  1/2

P(B/A)  =  0.5

(ii)  If a new oil filter is needed, what is the probability that the oil has to be changed?

The event A depends on B.

P(A/B)  =  P(AnB)/P(B)

=  0.15 / 0.40

=  3/8

P(A/B)  =  0.375

After having gone through the stuff given above, we hope that the students would have understood, "Practice Questions on Conditional Probability"

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