---
title: "Law of addition; satisfactory trainee Bayes"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:law-addition-satisfactory-trainee"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/law-addition-satisfactory-trainee
site: BCT Engineering
---

# Law of addition; satisfactory trainee Bayes

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

- Asked in: 2072 Kartik, 2+4 marks
- Video: [Bayes' Theorem EXPLAINED with Examples](https://www.youtube.com/watch?v=cqTwHnNbc8g) — Ace Tutors
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/law-addition-satisfactory-trainee
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

State the law of addition of probability. In a training, the 70% of persons achieved a rating of Satisfactory. Of those rated as Satisfactory, 80% had Acceptable Scores on the personality test. Of those rated as Unsatisfactory, 35% had Acceptable Scores. Find the probability that an applicant would be a Satisfactory trainee given the Acceptable scores on personality test.

## Short answer

For any events A,B, P(A ∪ B) = P(A)+P(B)−P(A∩B); if mutually exclusive, P(A∪B)=P(A)+P(B). Here P(acceptable) = 0.665, so P(satisfactory | acceptable) = 0.56 / 0.665 ≈ 0.842.

## Answer

#### Law of addition of probability
For any two events A and B:

```text
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
```

If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

#### Trainee problem
Let S be "rated Satisfactory" and T be "Acceptable score on the personality test".

```text
P(S) = 0.70,   P(S′) = 0.30
P(T|S) = 0.80, P(T|S′) = 0.35
```

Probability of an acceptable score (total probability):

```text
P(T) = P(S) P(T|S) + P(S′) P(T|S′)
     = (0.70)(0.80) + (0.30)(0.35)
     = 0.560 + 0.105
     = 0.665
```

Probability of a Satisfactory trainee given an acceptable score (Bayes' theorem):

```text
P(S|T) = P(S) P(T|S) / P(T) = 0.560 / 0.665 = 0.842
```

**Answer:** P(Satisfactory | Acceptable score) ≈ 0.842.

## Check yourself

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

1. What is the general law of addition for two events?
   A. P(A or B) = P(A) - P(B)
   B. P(A or B) = P(A) + P(B) - P(A and B)
   C. P(A or B) = P(A) x P(B)
   D. P(A or B) = P(A) + P(B) + P(A and B)
   Answer: B. The overlap is counted twice when adding, so it is subtracted. For mutually exclusive events it is zero. (Key point: Addition law)

2. What is the probability of an acceptable score?
   A. 0.665
   B. 0.35
   C. 0.560
   D. 0.80
   Answer: A. P(T) = 0.70x0.80 + 0.30x0.35 = 0.560 + 0.105 = 0.665. (Key point: Total probability)

3. Which calculation gives P(Satisfactory | Acceptable)?
   A. 0.105/0.665
   B. 0.665/0.560
   C. 0.80 x 0.70 x 0.35
   D. 0.560/0.665
   Answer: D. By Bayes' theorem it is P(S)P(T|S)/P(T) = 0.560/0.665 = 0.842. (Key point: Bayes)

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