**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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