---
title: "Mutually exclusive vs equally likely; Bayes topper"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:bayes-top-student-electronics"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/bayes-top-student-electronics
site: BCT Engineering
---

# Mutually exclusive vs equally likely; Bayes topper

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

- Asked in: 2070 Asar, 6 marks
- Video: [Introduction To Mutually Exclusive, Exhaustive And Equally Likely Events / Maths Probability](https://www.youtube.com/watch?v=RrQqHAQLDoE) — We Teach Academy Maths
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/bayes-top-student-electronics
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

Distinguish between mutually exclusive and equally likely events with examples. What is the use of Bayes theorem in theory of probability? In a college 45% students belong to Civil, 30% Electronics and remaining to other faculties. The probability of being top is 5%, 4% and 2% respectively. If this year's result is published, what is the probability that the topper is from electronics?

## Short answer

Mutually exclusive events cannot occur together (P(A∩B)=0); equally likely events have equal probability. Bayes' theorem updates cause probabilities after observing an effect. For the college: P(Electronics|Top) = (0.30×0.04) / (0.45×0.05 + 0.30×0.04 + 0.25×0.02) = 0.304.

## Answer

#### Mutually exclusive vs equally likely events

| Basis | Mutually exclusive | Equally likely |
|---|---|---|
| Meaning | Cannot occur together in one trial | Each has the same chance of occurring |
| Condition | P(A ∩ B) = 0 | P(A) = P(B) |
| Example | Head and tail in a single toss | Getting any one of the faces 1 to 6 on a fair die |
| Relation | Events may or may not be equally likely | Events may or may not be mutually exclusive |

#### Use of Bayes' theorem
Bayes' theorem revises the probability of a cause after an outcome has been observed. It is used for diagnosis (which cause produced the effect), such as finding which machine made a defective item, medical testing, spam filtering and decision making with new evidence.

#### Topper problem
Faculty shares: Civil 45%, Electronics 30%, others 100 − 45 − 30 = 25%. Let T be "student is the topper".

| Faculty | P(faculty) | P(T given faculty) | Product |
|---|---|---|---|
| Civil | 0.45 | 0.05 | 0.0225 |
| Electronics | 0.30 | 0.04 | 0.0120 |
| Others | 0.25 | 0.02 | 0.0050 |
| Total | | | 0.0395 |

```text
P(Electronics | T) = 0.0120 / 0.0395 = 0.3038
```

**Answer:** the probability that the topper is from Electronics is about 0.304.

## Check yourself

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

1. What is Bayes' theorem used for?
   A. To list the sample space
   B. To find the median
   C. To compute the mean
   D. To find the cause after an outcome is observed
   Answer: D. It revises the probability of a cause given an outcome, e.g. diagnosis. (Key point: Use of Bayes)

2. What share of students is in other faculties?
   A. 45%
   B. 25%
   C. 75%
   D. 30%
   Answer: B. 100 - 45 - 30 = 25%. (Key point: Other faculties)

3. What is P(Electronics | topper)?
   A. 0.04
   B. 0.30
   C. 0.304
   D. 0.012
   Answer: C. 0.0120/0.0395 = 0.3038. (Key point: Posterior)

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