- 2070 Asar · 6 marks
Mutually exclusive vs equally likely; Bayes topper
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.
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?
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 |
P(Electronics | T) = 0.0120 / 0.0395 = 0.3038
Answer: the probability that the topper is from Electronics is about 0.304.
✓ Answer checked by Claude Sonnet 5.5
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