Question : Give an example of Baye theorem of probability.
Answer :
Baye’s Theorem of probability : If P(A) and P(B/A),P(B/A^c) are known, then
\[P\left(\frac{A}{B}\right)\ = \frac{P\left(\frac{B}{A}\right)P(A)}{P\left(\frac{B}{A}\right)P\left(A\right)+P\left(\frac{B}{A^c}\right)P(A^c)}\]
Example of Baye theorem : A bag contains 2 white and 3 red shirts and a bag B contains 4 white and 5 red shirts. One shirt is drawn at random from one of the bags and it is found to be red. Find the probability that the red shirt drawn Is drawn from bag B.
Solution:
There are two bags.
Probability of selecting Bag A =P(A) =1/2
Probability of selecting Bag B =P(B) =1/2
Probability of selecting a red shirt from bag A = P(R/A)= red shirt in bag A / total shirt of bags A = 3/5
Probability of selecting a red shirt from bag B = P(R/B)= red shirt in bag B / total red shirt of bags A = 5/9
Probability of selecting bag B to draw a red shirt = P(B/R)
By Baye’s theorem
\[P\left(\frac{B}{R}\right)\ =\ \frac{P\left(\frac{R}{B}\right)P(B)}{P\left(\frac{R}{B}\right)P\left(B\right)+P\left(\frac{R}{A}\right)P(A)} \] \[P\left(\frac{B}{R}\right)\ =\ \frac{\frac{5}{9}\frac{1}{2}}{\frac{3}{5}\frac{1}{2}+\frac{5}{9}\frac{1}{2}}=\frac{\frac{5}{9}}{\frac{52}{45}}=\frac{25}{52}\]