During my school times, I found Bayes’ Theorem from the chapter on probability very tough. Do you find it tough too? Do not worry. We will help you understand what this theorem is and where it can be applied in practical life. Bayes’ theorem was formulated by a British statistician named Thomas Bayes in the 18th century. Let us understand the need for Bayes’ theorem. Consider that there are two boxes, box A and box B. Box A contains 2 black balls and 3 white balls and box B contains 4 black balls and 5 red balls. Suppose one ball is drawn at random from one of the boxes. We can find the probability of selecting any one of the bags (i.e., ½) or the probability of drawing a ball of a particular colour from a particular box if we are given the box from which the ball is drawn. But can we find the probability that the box is drawn is from a particular box if the colour of the ball is known to us? Here, we have to find the reverse probability of any box to be selected when an event occurred after it is known. Here is where the role of Bayes’ theorem comes into play.
Bayes’ Theorem in Probability
As we all know, probability is the part of mathematics that is concerned with how likely an event is to occur, or how likely it is that a proposition is true. In probability theory, Bayes’ theorem is used to determine reverse probability by using conditional probability. Conditional probability is the probability of an outcome occurring, based on a previous outcome occurring. The formula for conditional probability is given below:
P(X/Y) = P(X∩Y)/ P(Y)
P(X/Y) means Probability of event X, given Y has occurred,
P(X∩Y) means Probability that event X and event Y occur together and
P(Y) means Probability of event Y.
Formula for Bayes’ Theorem
The Bayes’ theorem is stated as below:
P(X/Y) = P(Y/X) P(X)/ P(Y)
P(X/Y) = the probability of event X occurring, given event Y has occurred.
P(Y/X) = the probability of event Y occurring, given event X has occurred.
P(X) = the probability of event X
P(Y) = the probability of event Y
It is important for us to note that events X and Y are independent events (i.e., the probability of the outcome of event X does not depend on the probability of the outcome of event Y).
Example of Bayes’ Theorem
Let us suppose that you are an analyst at an investment bank and according to your survey 60% of the large-cap companies that increased their stock price by more than 5% in the last three years replaced their CFOs during the period.
Concurrently, only 35% of the large-cap companies that did not increase their share price by more than 5% in the same period replaced their CFOs. Knowing that the probability that the stock prices grow by more than 5% is 4%, we need to find the probability that the shares of a large-cap company that fires its CFO will increase by more than 5%.
Let us know the required probability. Let
P(X) = probability that the stock price rises by 5%
P(Y) = probability that the CFO is replaced
P(A/B) = probability that the stock price rises by 5% given that the CFO has been replaced.
P(B/A) = probability of the CFO replacement given the stock price has increased by 5%.
Using the Bayes’ theorem, we get:
P(A/B) = 0.60 × 0.04 / 0.60 × 0.04 + 0.35 × (1-0.04) = 0.067 or 6.67%.
Therefore, the probability that the shares of a large-cap company that replaces its CFO will grow by more than 5% is 6.67%.
Bayes’ theorem is an integral part of probability. If you want to learn more about these concepts in detail and in an interesting and fun way, visit Cuemath.