# Unconditional probability using the total probability rule

We have already discussed the calculation of the unconditional probability of an event using the total probability rules.

If the event A is dependent on event A_{1}, A_{2},...., A_{n} where A_{1}, A_{2},...., A_{n} are mutually exclusive and exhaustive events then according to the total probability rule:

P(A) = P(AA_{1}) + P(AA_{2}) + P(AA_{3}) + . .. ..+ P(AA_{n})

because A will always occur with one of A_{1}, A_{2},...., A_{n}

P(A) = P(A|A_{1})*P(A_{1}) + P(A|A_{2})*P(A_{2}) + .... + P(A|A_{n})*P(A_{n})

because P(AA_{1}) = P(A|A_{1})*P(A_{1})

So, the unconditional probability of an event is equal to the sum of joint probabilities of the event with mutually exclusive and exhaustive events.

The unconditional probability of an event A is equal to the sum of the product of conditional probabilities of event A with different mutually exclusive and exhaustive events and the probabilities of those events.

P(A) = Conditional probability of event A with event A_{1}*Probability of event A_{1 }+ ..... + Conditional probability of event A with event A_{n}*Probability of event A_{n}

where A_{1}, A_{2},...., A_{n} are mutually exclusive and independent events.

Example 2: Calculation of unconditional probability |
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The probability of an increase in sales of a company (based in the USA) by more than 20 percent is dependent on three mutually exclusive and exhaustive events. Those three events are- increase in overall GDP of the USA less than 5 percent, increase in GDP by more than or equal to 5 percent and less than 7 percent, and increase in GDP by more than or equal to 7 percent. The conditional and unconditional probabilities of the events are given below:
Compute the unconditional probability of an increase in sales of the company by more than 20 percent. Let us first name the events: Event A= Increase in sales of the company by more than 20 percent Event A Event A Event A We are given the following probabilities: P(A P(A|A Applying the total probability rule, we get the unconditional probability of event A: P(A) = P(A|A So, the unconditional probability of an increase in the sales of the company by more than 20 percent is |

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