Conditional Probability: Bayes' Theorem

Bayes’ theorem describes the probability of occurrence of an event related to any condition. For example: if we have to calculate the probability of taking a blue ball from the second bag out of three different bags of balls, where each bag contains three different color balls viz. red, blue, black. Such case where probability of occurrence of an event is calculated depending on other conditions is known as conditional probability.

Conditional Probability

Derivation of Bayes Theorem:

Statement:Let \(E_1, E_2,…,E_n\)} be a set of events associated with a sample space \(S\), where all the events \(E_1, E_2,…, E_n\) have nonzero probability of occurrence and they form a partition of \(S\). Let \(A\) be any event associated with \(S\), then according to Bayes theorem,

\(P(E_i│A)~=~\frac{P(E_i)P(E_i│A)}{\sum\limits_{k=0}^{n}P(E_k)P(A| E_k)}\)

Proof:According to conditional probability formula,

\(P(E_i│A)~=~\frac{P(E_i ∩ A)}{P(A)}\) ⋯⋯⋯⋯⋯⋯⋯⋯(1)

Using multiplication rule of probability,
\(P(E_i ∩ A)~= ~P(E_i)P(E_i│A)\)⋯⋯⋯⋯⋯⋯⋯⋯(2)

Using total probability theorem,
\(P(A)~=~\sum\limits_{k=0}^{n}~P(E_k)P(A| E_k)\)⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯(3)

Putting the values from equations (2) and (3) in equation 1, we get

\(P(E_i│A)~=~\frac{P(E_i)P(E_i│A)}{\sum\limits_{k=0}^n~P(E_k)P(A| E_k)}\)

Some illustrations will improve the understanding of the concept.

Illustration 1:Bag I contains 4 white and 6 black balls while another Bag II contains 4 white and 3 black balls. One ball is drawn at random from one of the bags and it is found to be black. Find the probability that it was drawn from Bag I.

Solution:Let \(E_1\) be the event of choosing the bag I, \(E_2\) the event of choosing the bag II and A be the event of drawing a black ball.

Then,\(P(E_1)~ = ~P(E_2)~ =~\frac{1}{2}\)

Also,\(P(A|E_1) ~= ~P\)(drawing a black ball from Bag I) = \(\frac{6}{10}~ = ~\frac{3}{5}\)

\(P(A|E_2) ~=~ P\)(drawing a black ball from Bag II) = \(\frac{3}{7}\)

By using Bayes’ theorem, the probability of drawing a black ball from bag I out of two bags,

\(P(E_1 |A)~ =~\frac{P(E_1)P(A|E_1)}{P(E_1 )P(A│E_1 )+ P(E_2)P(A|E_2)}\)

=\(\large\frac{\frac{1}{2}~\times~\frac{3}{5}}{\frac{1}{2}~\times~\frac{3}{7}~+~\frac{1}{2}~ ×~\frac{3}{5}}\) = \(\frac{7}{12}\)

Illustration 2:A man is known to speak truth 2 out of 3 times. He throws a die andreports that number obtained is a four. Find the probability that the number obtained is actually a four.

Solution:Let \(A\) be the event that the man reports that number four is obtained.

Let \(E_1\) be the event that four is obtained and \(E_2\) be its complementary event.

Then, \(P(E_1)\) = Probability that four occurs = \(\frac{1}{6}\)

\(P(E_2)\) = Probability that four does not occurs = \(1 ~–~ P(E_1) ~=~ 1~-\frac{1}{6}~ =~\frac{5}{6}\)

Also, \(P(A|E_1)\) = Probability that man reports four and it is actually a four = \(\frac{2}{3}\)

\(P(A|E_2)\) = Probability that man reports four and it is not a four = \(\frac{1}{3}\)

By using Bayes’ theorem, probability that number obtained is actually a four,

\(P(E_1 |A)~ \) \(= \large \frac{P(E_1)P(A|E_1)}{P(E_1 )P(A│E_1 )~+~ P(E_2)P(A|E_2)}~
=~\frac{\frac{1}{6} ~ ×~ \frac{2}{3}}{\frac{1}{6} ~×~ \frac{2}{3}~ +~ \frac{5}{6}~ ×~\frac{1}{3}}\)
= \(\frac{2}{7}\)<

For detailed discussion on the concept of Bayes’ theorem, download Byju’s-the learning app.

Practise This Question

A bag contains some white and some black balls, all combinations of balls being equally likely. The total number of balls in the bag is 10. If three balls are drawn at random without replacement and all of them are found to be black, the probability that the bag contains 1 white and 9 black balls is