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Maharashtra State BoardHSC Science (Computer Science) 11th Standard
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Bayes’ Theorem

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Estimated time: 7 minutes
CBSE: Class 12

Introduction

Bayes' Theorem is used to find the probability of a cause when the result is already known. It is often called a method of finding reverse probability, because it helps move from an observed event to the event that may have caused it.

CBSE: Class 12
Maharashtra State Board: Class 12

Theorem: Bayes' Theorem

If B1, B2,..., Bn are mutually exclusive and exhaustive events and if A is an event consequent to these Bi's, then for each i = 1, 2, 3, ..., n,

\[P(B_i | A) = \frac{P(B_i)P(A | B_i)}{\sum_{i=1}^{n} P(B_i)P(A | B_i)}\]
CBSE: Class 12

Example 1

A doctor may travel by train, bus, scooter, or other means. The probabilities of these are \[\frac{3}{10}, \frac{1}{5}, \frac{1}{10}\], and \[\frac{2}{5}\] respectively. The probabilities that he will be late are \[\frac{1}{4}, \frac{1}{3}\], and \[\frac{1}{12}\] if he comes by train, bus, and scooter respectively, while he is not late if he comes by other means. If he arrives late, find the probability that he came by train.

Step 1: Define the events

Let:

  • \[T_1 =\] comes by train.

  • \[T_2 =\] comes by bus.

  • \[T_3 =\] comes by scooter.

  • \[T_4 =\] comes by other means.

  • \[E =\] doctor arrives late.

Step 2: Write the probabilities

  • \[P(T_1) = \frac{3}{10}, P(T_2) = \frac{1}{5}, P(T_3) = \frac{1}{10}, P(T_4) = \frac{2}{5}\].

  • \[P(E | T_1) = \frac{1}{4}, P(E | T_2) = \frac{1}{3}, P(E | T_3) = \frac{1}{12}, P(E | T_4) = 0\].

Step 3: Apply Bayes' Theorem

\[P(T_1 | E) = \frac{P(T_1)P(E | T_1)}{P(T_1)P(E | T_1) + P(T_2)P(E | T_2) + P(T_3)P(E | T_3) + P(T_4)P(E | T_4)}\]

Substituting:

\[P(T_1 | E) = \frac{\frac{3}{10} \cdot \frac{1}{4}}{\frac{3}{10} \cdot \frac{1}{4} + \frac{1}{5} \cdot \frac{1}{3} + \frac{1}{10} \cdot \frac{1}{12} + \frac{2}{5} \cdot 0}\]
 
\[P(T_1 | E) = \frac{1}{2}\]
CBSE: Class 12

Key Points: Bayes' Theorem

  • Bayes' Theorem works from effect to cause.

  • Prior probability means “before observation”.

  • Posterior probability means “after observation”.

  • The observed event is usually given in the question statement.

  • The required answer is often a probability of the form P(cause ∣ observed event).

Video Tutorials

We have provided more than 1 series of video tutorials for some topics to help you get a better understanding of the topic.

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Series 4


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