Let S be the Sample space and let F

Event F _{1} , F_{2}……F_{n} |
They are called hypothesis |

P(F _{1}) …… P(F_{2}) |
These probabilities are called priori probabilities as they exist before we obtain any information from experiment |

P( E/F _{1}) ……. P(E/F_{n}) |
These probabilities are called likelihood probabilities as they tell us how event A under consideration occurs |

P(F _{1}/E) …. P(F_{n}/E) |
These probabilities are called the posterior probabilities as they are determined after the result of the experiment is known |

Examples:

**Question**

Of all the smokers in a particular district in India 40% prefer brand X and 60% prefer brand Y. Of those smokers who prefer brand X, 30% are females, and of those who prefer brand Y, 40% are female. What is the probability that a randomly selected smoker prefers brand A, given that the person selected is a female?

**Solution**

Let F_{1} = "prefer brand A",

F_{2} = "prefer brand B" and

E is the event "female".

We need to find out P(F_{1}|E)

Now from Bayes formula

In this particular situation

So

Or

P(F_{1}|E) =1/3

Of all the smokers in a particular district in India 40% prefer brand X and 60% prefer brand Y. Of those smokers who prefer brand X, 30% are females, and of those who prefer brand Y, 40% are female. What is the probability that a randomly selected smoker prefers brand A, given that the person selected is a female?

Let F

F

E is the event "female".

We need to find out P(F

Now from Bayes formula

In this particular situation

So

Or

P(F

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