Unconditional and Conditional Probabilities

Let’s say you are asked the following question:

What is the probability of your portfolio earning a return greater than 10%?

This kind of probability is an unconditional probability as the probability is not dependent on the occurrence of any other event. The event, A, is that the portfolio will earn a return greater than 10%. The probability of such an event will be specified as P(A). The calculation is quite simple. The numerator is the sum of probabilities of all returns being above 10%. Assume this is 0.60. The denominator is 1, the sum of probabilities of all possible returns. The probability P(A) = 0.60/1 = 0.60.

Now, let’s ask another related question:

What is the probability of your portfolio earning a return greater than 10% given that the returns are never below 5%?

Notice that we have added a new condition - given that the returns are never below 5%. Now the probability of portfolio earning returns greater than 10% is not unconditional. It is conditional on another event, B, that is, the returns are never below 5%. Such a probability is called conditional probability, and is expresses as P(A|B), the probability of A given B.

Our calculation will now change.

The numerator will still be the same: the sum of probabilities of all returns being above 10%. We assumed this to be 0.60.

The denominator will now consider Event B as well - the sum of probabilities of all returns being 5% or more. Assume this is 0.80.

The conditional probability will be calculated as P(A|B) = 0.60/0/80 = 0.75.