A conditional probability is expressed as

P(A | B)

and read as the "probability of (event) A (will happen) given (that event) B (has happened).".

In short form, the "probability of A given B".The vertical bar "|" is read as "given that".

What is the probability it rains today given that it rained yesterday?

What is the probability you get an "A" on this exam given that you got an "A" on the last exam?

Notice the time element of probability. We are assuming that event B has happened and want to know about the probability of event A happening.

To make the concepts easier to understand, the time element is used.

Strictly speaking, one can do without the time element, but the concepts can become more difficult to understand and/or visualize.

Event A is some event.

There is a probability, P(A) that event A has happened.

Event B is another event.

There is a probability, P(B) that event B has happened.

If the events A and B are mutually exclusive, then they cannot happen together.

P(A ∩ B) = 0.0
P(A & B) = 0.0

If the events A and B are mutually exclusive, then they cannot happen together.

P(A ∪ B) = 1.0
P(A | B) = 1.0

That is, together they cover every possible event in the universe of possible events.

The intersection of events A and B is the joint probability.

Conditional probability is concerned with events that can happen together.

Otherwise, the conditional probability is 0.0 (zero).

Assume that event B has happened. What is the probability that event A has happened? From the diagram, one can see that it is the following.

P(A | B) = P(A & B) / P(B)

That is, we ignore the part of A that does not intersect with B.

Rearrange to get the following.

P(A & B) = P(A | B) * P(B)

Assume that event A has happened. What is the probability that event B has happened? From the diagram, one can see that it is the following.

P(B | A) = P(A & B) / P(A)

That is, we ignore the part of B that does not intersect with A.

Rearrange to get the following.

P(B & A) = P(B | A) * P(A)

To use these rules for problem solving, remember that there are 4 unknowns, so that given 3 of the unknowns, you can solve for the remaining unknown.

P(A & B) = P(A | B) * P(B)
P(B & A) = P(B | A) * P(A)

Since

P(A & B) = P(B & A)

it follows that

P(A | B) * P(B) = P(B | A) * P(A)

which is Bayes Rule usually written as follows.

P(A | B) = P(B | A) * P(A) / P(B)

One each of two sides of three cards, one card is red-red, one is red-black, and one is black-black. If you pick one of the three cards at random and the side facing you is red, what is the probability that the other side is red. Justify your answer.

Assume that the probability of any one person carrying a bomb onto a commercial airliner is 0.0000001. You decide to carry a bomb onto such a plane (you are not caught in your attempt). What is the probability that two people have each carried bombs onto the plane? What is the probability that at least one person has a bomb on the plane?

A radio station picks a number from 1 to 100. Callers then attempt to guess the number. The first caller to guess the number wins a pair of tickets to an upcoming concert. Obviously, no number should be guessed if it is already known to be incorrect. Does it matter what number caller you are as far as your chances of guessing the number? Explain.

Two events are said to be independent if the outcome of each does not depend on the other. Otherwise, the events are said to be dependent.

We know from the multiplication rule (a variant of the conditional probability rule) that

P(A | B) = P(A & B) / P(B)
P(B | A) = P(A & B) / P(A)

If two events are independent, then the following hold.

P(A | B) = P(A)
P(B | A) = P(B)

That is:

• The probability of A occurring is not dependent on B occurring.
• The probability of B occurring is not dependent on A occurring.

The next question is: What is the nature of the dependence?

From the development of the conditional probabilities, the following hold.

P(B & A)
= P(A & B)
= P(B | A) * P(A)
= P(A | B) * P(B)

From the last two expressions, one gets Bayes Rule. Because of this simple development, it has been rediscovered and used often in the past few hundred years. The property of independence goes the other way, in that if the multiplication law holds, then the events must be independent.

Independence is often determined by intuition and not by solving equations and doing number crunching. One application is in experimental testing. If random samples are taken, then the dependence of one event on another event is minimized (effectively nil) and so the shortened form of the multiplication rule applies.

Consider employee performance. If the probability that a customer complains about an employee is 0.02 (1 of every 50 customers) then what is the probability that a customer complains about an employee given that another customer has complained about that same employee? That is

P(complaint | previous complaint)

If the probability is still 0.02, then the events are independent. But if the probability is greater (or smaller), then the events are not independent. Remember that two events A and B are said to be mutually exclusive if they have no sample points in common. That is, the following holds.

P(A ∩ B) = 0.0

If one of two possible mutually exclusive events is known to occur, the other is then known not to occur (since, by definition, one of the two events must occur).

If the two events were independent, then there is no connection between the two events.

So, two events with nonzero probabilities cannot be both mutually exclusive and independent.