**Practice Questions on Conditional Probability :**

Here we are going to see some practice questions on conditional probability.

**Question 1 :**

Can two events be mutually exclusive and independent simultaneously?

**Solution :**

If two events A and B are mutually exclusive then

P(AnB) = 0 ----(1)

If two events A and B are independent then

P(AnB) = P(A) ⋅ P(B) ----(2)

(1) = (2)

P(A) ⋅ P(B) = 0

**Case 1 :**

If either P(A) or P(B) be equal to 0, then the events will be mutually exclusive or independent.

**Case 2 : **

If P(A) > 0 and P(B) > 0, then

P(AnB) = P(A) P(B) > 0 (Which is not equal to 0)

So, the events are independent not mutually exclusive.

**Case 3 : **

If P(A) > 0 and P(B) = 0 or P(A) = 0 and P(B) > 0, then

P(AnB) = P(A) P(B) = 0

So, the events are mutually not independent exclusive.

Hence, we may conclude that two events cannot be mutually exclusive and independent simultaneously.

**Question 2 :**

If A and B are two events such that P(A U B) = 0.7, P(A n B) = 0.2, and P(B) = 0.5, then show that A and B are independent.

**Solution :**

To show that event A and B are independent, we have to show that P(AnB) = P(A) ⋅ P(B)

P(AUB) = P(A) + P(B) - P(AnB)

0.7 = P(A) + 0.5 - 0.2

0.7 = P(A) + 0.3

0.7 - 0.3 = P(A)

P(A) = 0.4

P(AnB) = P(A) ⋅ P(B)

0.2 = 0.4 (0.5)

0.2 = 0.2

Hence the events A and B are independent.

**Question 3 :**

If A and B are two independent events such that P(A∪B) = 0.6, P(A) = 0.2, find P(B).

**Solution :**

P(A∪B) = 0.6, P(A) = 0.2

Since A and B are independent events, P(AnB) = 0

P(AnB) = P(A) + P(B) - P(AUB)

0 = 0.2 + P(B) - 0.6

0 = -0.4 + P(B)

P(B) = 0.4

**Question 4 :**

If P(A) = 0.5, P(B) = 0.8 and P(B/A) = 0.8, find P(A / B) and P(A∪B) .

**Solution :**

P(A/B) = P(AnB)/P(B) ----(1)

P(B/A) = P(AnB)/P(A)

0.8 = P(AnB)/0.5

0.8(0.5) = P(AnB)

P(AnB) = 0.4

By applying the value of P(AnB) in (1), we get

P(A/B) = 0.4 / (0.8)

P(A/B) = 0.5

P(AUB) = P(A) + P(B) - P(AnB)

P(AUB) = 0.5 + 0.8 - 0.4

P(AUB) = 0.9

**Question 5 :**

If for two events A and B, P(A) = 3/4, P(B) = 2/5 and AUB = S (sample space), find the conditional probability P(A/B)

**Solution :**

P(A) = 3/4, P(B) = 2/5 and AUB = S

P(AUB) = P(A) + P(B) - P(AnB)

1 = (3/4) + (2/5) - P(AnB)

1 = (15 + 8)/20 - P(AnB)

1 - (23/20) = P(AnB)

P(AnB) = 3/20

P(A/B) = P(AnB)/P(B)

= (3/20) / (2/5)

= (3/20) ⋅ (5/2)

P(A/B) = 3/8

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