# Solved Problems On Bayes Theorem

Union of Events – The union of two or more events is basically the combined set of the two or more events in the sample space.For example, the union of getting a king or hearts in a deck of cards would include 16 cards (13 hearts and 3 kings).

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Also, this theorem can be used in different ways to obtain interesting and meaningful results.

Bayesian inference is a method of statistical inference in which Bayes’ theorem is used to update the probability of a hypothesis as more evidence or information becomes available.

Note that we do not count the king of hearts twice.

Intersection of Events – The intersection of two or more events is the set of events that are common in both the events.

Event – An event is simply the outcome of an experiment.

For example, if you pick out a card and get a queen of spades, it is an event.This very simple example helps us understand the Bayes rule for an uncomplicated case.This theorem can be extended to many cases which are more complicated.Let us understand this theorem with the help of a simple example. There are two shapes of chocolate – round and flat. We would like to find out the probability of getting a round chocolate given that the we have already got a round chocolate out.This can be solved by using the Bayes’ theorem in the following way: P(1) = Probability of getting a round chocolate at the first pick = 7/15 P(2) = Probability of getting a round chocolate at second pick = 6/14 P(1 and 2) = ½*(6/14) P(2/1) = P(1 and 2)/P(1) = 0.214/0.4667=0.4585So the probability of getting a round chocolate on the second picking given that the first picking yielded a round chocolate is 45.85%.Sample Space – The sample space is the collection of all possible outcomes.Taking the previous example of cards, the sample space would be all 52 cards.This is the initial probability that the light bulb is on. This is the evidence that alters p(i) to a more accurate value.Now, you try to gather evidence that will tell you if the light bulb is on or not. So we finally get p(i/switch) that is the probability of light being on given that the switch is on.The methods related to using Bayes’ theorem come under Bayesian Statistics where various methods like Bayesian inference, Probability trees, Bayes’ updating and Frequentist approach.Aman Goel is a Computer Science and Engineering graduate from IIT Bombay. He is now the co-founder of Allin Call Research & Solutions.