---
title: "Multiplication law; applicant passing test"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:law-multiplication-electronics-applicant"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/law-multiplication-electronics-applicant
site: BCT Engineering
---

# Multiplication law; applicant passing test

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

- Asked in: 2073 Shrawan, 6 marks
- Video: [Multiplication & Addition Rule - Probability - Mutually Exclusive & Independent Events](https://www.youtube.com/watch?v=94AmzeR9n2w) — The Organic Chemistry Tutor
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/law-multiplication-electronics-applicant
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

State the law of multiplication of probability. An Electronics company has an engineering position open. The Probability that an applicant is capable is 0.7. Each applicant is given written test and oral examination. A capable applicant passes with Probability 0.9 while an incapable applicant passes with Probability of 0.4. Find (a) the probability that an applicant passes the test (b) the probability that the applicant is capable given he/she passes the test.

## Short answer

For any events A and B, P(A ∩ B) = P(A) P(B|A). Using total probability, P(pass) = 0.7×0.9 + 0.3×0.4 = 0.75. By Bayes, P(capable | pass) = 0.63 / 0.75 = 0.84.

## Answer

#### Law of multiplication of probability
For two events A and B, the probability that both occur is

```text
P(A ∩ B) = P(A) · P(B | A) = P(B) · P(A | B)
```

If A and B are independent, P(B | A) = P(B), so P(A ∩ B) = P(A) · P(B). For three events, P(A ∩ B ∩ C) = P(A) P(B|A) P(C|A ∩ B).

#### Applicant problem
Let C be "applicant is capable" and T be "applicant passes".

```text
P(C) = 0.7,   P(C′) = 0.3
P(T|C) = 0.9, P(T|C′) = 0.4
```

**(a) Probability that an applicant passes**

```text
P(T) = P(C) P(T|C) + P(C′) P(T|C′)
     = (0.7)(0.9) + (0.3)(0.4)
     = 0.63 + 0.12
     = 0.75
```

**(b) Probability of being capable given the applicant passes**

```text
P(C|T) = P(C) P(T|C) / P(T) = 0.63 / 0.75 = 0.84
```

**Answer:** (a) 0.75, (b) 0.84.

## Check yourself

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

1. What does the law of multiplication say for two events A and B?
   A. P(A and B) = P(A) - P(B)
   B. P(A and B) = P(A) x P(B given A)
   C. P(A and B) = P(A) + P(B)
   D. P(A and B) = P(A)/P(B)
   Answer: B. The probability that both occur is P(A) times the conditional probability of B given A; for independent events P(B|A) = P(B). (Key point: Multiplication law)

2. How is the probability that an applicant passes found?
   A. 0.7 + 0.9
   B. P(C)P(T|C) + P(C')P(T|C') = 0.63 + 0.12
   C. 0.9 x 0.4
   D. 0.7 x 0.4
   Answer: B. Sum over capable and incapable applicants: 0.7x0.9 + 0.3x0.4 = 0.75. (Key point: Total probability)

3. What is the probability that a passing applicant is capable?
   A. 0.84
   B. 0.90
   C. 0.70
   D. 0.63
   Answer: A. P(C|T) = 0.63/0.75 = 0.84. The 0.63 is P(C and T) and 0.75 is P(T). (Key point: Bayes)

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