JEE Main & Advanced Mathematics Probability Total Probability and Baye's Rule

(1) The law of total probability : Let S be the sample space and let \[{{E}_{1}},\,{{E}_{2}},\,.....{{E}_{n}}\] be n mutually exclusive and exhaustive events associated with a random experiment. If A is any event which occurs with \[{{E}_{1}}\] or \[{{E}_{2}}\] or …or  \[{{E}_{n}},\] then

\[P(A)=P({{E}_{1}})\,P(A/{{E}_{1}})+P({{E}_{2}})\,P(A/{{E}_{2}})+...+P({{E}_{n}})\,P(A/{{E}_{n}})\].

(2) Baye’s rule : Let S be a sample space and \[{{E}_{1}},\,{{E}_{2}},\,.....{{E}_{n}}\] be n mutually exclusive events such that \[\bigcup\limits_{i=1}^{n}{{{E}_{i}}}=S\] and \[P({{E}_{i}})>0\] for \[i=\text{ }1,\text{ }2,\text{ }\ldots \ldots ,n\]. We can think of (\[{{E}_{i}}'s\] as the causes that lead to the outcome of an experiment. The probabilities \[P({{E}_{j}}),\,\,i=1,\,\,2,\,\,...,\,n\]  are called  prior probabilities. Suppose the experiment results in an outcome of event A, where \[P(A)>0\]. We have to find the probability that the observed event A was due to cause \[{{E}_{i}},\] that is, we seek the conditional probability \[P({{E}_{i}}/A)\]. These probabilities are called posterior probabilities, given by Baye’s rule as \[P({{E}_{i}}/A)=\frac{P({{E}_{i}}).P(A/{{E}_{i}})}{\sum\limits_{k=1}^{n}{P({{E}_{k}})\,P(A/{{E}_{k}})}}\].