trogulj

Er... No offense guys.

I know everyones only trying to help here and has the noblest of intentions, but Sujan's explanation and example don't have anything to do with the total probability rule.

The total probability rule expresses unconditional probabilities in terms of conditional probabilities for mutually exclusive and exhaustive events.

Suppose that we are concerned with determining the probability of you carrying an umbrella on a given day, P(A).

We are given the following information:

The probability of rain today = P(S) = 0.4
The probability of no rain today = P(Sc) = 0.6
P(S) plus P(Sc) equals 1 and they are mutually exclusive, exhaustive events.

The probability of you carrying an umbrella, P(A), an unconditional probability, equals the sum of two joint probabilities:

P(carrying an umbrella and rain) ie. P(AS) ; plus P(carrying an umbrella and no rain) ie. P(ASc)

P(carrying an umbrella and rain) can be calculated as P(carrying umbrella given rain) * P(rain)

Similarly:
P(carrying an umbrella and no rain) can be calculated as P(carrying umbrella given no rain) * P(No rain)

Therefore, if you are also provided the following information:

The probability of you carrying an umbrella given that it rains = P(A|S) = 0.85
The probability of you carrying an umbrella given that it does not rain = P(A|Sc) = 0.25

You can calculate the probability of carrying an umbrella as:

P(A) = P(AS) + P(ASc)
P(A) = P(A|S) * P(S) + P(A|Sc) * P(Sc)
P(A) = (0.85*0.4) + (0.25*0.6) = 0.49

The total probability rule is a way of calculating an unconditional probability, P(A) using conditional probabilities, P(A|S) and P(A|Sc). The unconditional probabilities, P(A|S) and P(A|Sc), are basically used to calculate the joint probabilities, P(AS) and P(ASc). The sum of P(AS) and P(ASc) give you P(A) if S and Sc are mutually exclusive and exhaustive scenarios.

I HATE RESOLVING QUANT ISSUES