Name: Multiplication Theorem on Probability
Uploaded: 2019-07-19
Duration: 2 min 37 s
Description: Everything that happens in life is based on probability. Do you know what the multiplication theorem of probability is? Watch the video for a clear explanati...

Question

State and prove Bayes' theorem.

Accepted Answer

The statement of Bayes' theorem is as follows: Let $E_{1}, E_{2}, E_{3}, . . . . ., E_{n}$ be a set of events associated with a sample space $S$ , where all events $E_{1}, E_{2}, E_{3}, . . . . ., E_{n}$ have a non-zero probability of occurrence and they form a partition of $S$ . Let $A$ be any event which occurs with $E_{1} or E_{2} or E_{3} . . . . . or E_{n}$ , then according to Bayes Theorem,

$P (E_{i} / A) = \frac{P (E_{i}) P (A / E_{i})}{\sum_{k = 1}^{n} P (E_{k}) P (A / E_{k})}, i = 1, 2, 3, . . . ., n$

Here $E_{i} \cap E_{j} = ϕ$ , where $i \neq j$ , i.e. they are mutually exhaustive events.

The union of all the events of the partition should give the sample space.

To prove the Bayes' theorem, we will use the total probability and conditional probability formulas.

$E_{i}$ be is a partition of the sample space $S$ . Let $A$ be an event that occurred.

According to conditional probability, $P (E_{i} / A) = \frac{P (E_{i} \cap A)}{P (A)} . . . . . . . (1)$

Using the multiplication rule of probability, $P (E_{i} \cap A) = P (E_{i}) P (A / E_{i})$

Here, $P (A) = P (E_{1} \cap A) + P (E_{2} \cap A) + . . . . + P (E_{n} \cap A)$

$\Rightarrow P (A) = P (E_{1}) P (A / E_{1}) + P (E_{2}) P (A / E_{2}) + . . . . + P (E_{n}) P (A / E_{n})$

From the total probability theorem, $\Rightarrow P (A) = \sum_{k = 1}^{n} P (E_{k}) P (A / E_{k})$

Substituting values in equation $(1)$ ,

$P (E_{i} / A) = \frac{P (E_{i}) P (A / E_{i})}{\sum_{k = 1}^{n} P (E_{k}) P (A / E_{k})}, i = 1, 2, 3, . . ., n$ .

Hence, proved.

State and prove Bayes' theorem.

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