Introduction to Data Mining

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The joint probability distribution defines all the probability assignments to all possible propositions in a given domain.

A probabilistic model consists of a fixed set of random propositions, also called as random variables. An atomic event is an assignment of particular possible values to all of these random variables.

An joint probability distribution table lists out the probabilities for the occurrence of all possible combinations of values that could be assigned to all the random variables. The summation of all the probability values in the table should be always equal to 1.

Example: If we have only two random variables, cavity and toothache. Both of them are Boolean, i.e. they can have two possible values, yes or no. The joint probability distribution table could have the following assignments.

Answer:

P(Cavity) = 0.04+0.06         = 0.1

P(Toothache) = 0.04+0.01  = 0.05

P(Cavity ∩ Toothache)        = 0.04

P(Cavity U Toothache) = P(Cavity) + P(Toothache) - P(Cavity ∩ Toothache)

                                     = 0.1 + 0.05 - 0.04 = 0.11

Also,

P(Cavity U Toothache) = P(Cavity ∩ Toothache) + P(!Cavity ∩ Toothache) + P(Cavity ∩ !Toothache)

                                     = 0.04 + 0.06 + 0.01 = 0.11

1. For the given joint probability distribution, find the conditional probabilities P(Cavity/Toothache), P(Toothache/Cavity), P(Cavity/!Toothache), and P(Toothache/!Cavity).

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