---
title: "Find P(B) given P(A), P(A∪B)"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:p-b-independent-mutually-exclusive"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/p-b-independent-mutually-exclusive
site: BCT Engineering
---

# Find P(B) given P(A), P(A∪B)

Probability and Statistics · Chapter 1: Descriptive Statistics and Basic Probability

- Asked in: 2068 Jestha (old course), 6 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/p-b-independent-mutually-exclusive
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

Let P(A) = 0.4 and P(A∪B) = 0.7. What is P(B) if:
i) A and B are independent?
ii) A and B are mutually exclusive?

## Short answer

For independent events P(B)=0.5. For mutually exclusive events P(B)=0.3.

## Answer

Given: P(A) = 0.4 and P(A ∪ B) = 0.7. By the addition law, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

#### i) A and B independent
For independent events, P(A ∩ B) = P(A) P(B).

```text
0.7 = 0.4 + P(B) − 0.4 P(B)
0.7 − 0.4 = P(B) (1 − 0.4)
0.3 = 0.6 P(B)
P(B) = 0.5
```

Check: P(A ∩ B) = 0.4 × 0.5 = 0.2, and P(A ∪ B) = 0.4 + 0.5 − 0.2 = 0.7. ✓

#### ii) A and B mutually exclusive
For mutually exclusive events, P(A ∩ B) = 0.

```text
0.7 = 0.4 + P(B)
P(B) = 0.3
```

**Answer:** (i) P(B) = 0.5, (ii) P(B) = 0.3.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which law links P(A union B) with P(B)?
   A. P(A)P(B)
   B. P(A)-P(B)
   C. P(A)/P(B)
   D. P(A)+P(B)-P(A and B)
   Answer: D. Use P(A union B) = P(A) + P(B) - P(A and B). (Key point: Addition law)

2. What is P(B) when A and B are independent?
   A. 0.5
   B. 0.6
   C. 0.7
   D. 0.3
   Answer: A. 0.7 = 0.4 + P(B) - 0.4P(B) gives 0.6P(B) = 0.3, P(B) = 0.5. (Key point: Independent case)

3. What is P(B) when mutually exclusive?
   A. 1.1
   B. 0.3
   C. 0.28
   D. 0.5
   Answer: B. P(A and B) = 0, so P(B) = 0.7 - 0.4 = 0.3. (Key point: Exclusive case)

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