13D - Conditional probability

Conditional probability

Often we are interested in the probability of an event A occurring, given that event B has already occurred - this is conditional probability. Mathematically we denote the conditional probability as Pr(A|B) - this reads "the probability of event A occurring given event B has already occurred." In sum, conditional probability can be thought of as adjusting the probability to account for known information.

Mr. and Mrs. Smith

At the beginning of this chapter we introduced Mr. and Mrs. Smith who had two children, at least one of which was a girl. The question was then asked: What is the probability both of their children are girls? This is one example of conditional probability, we have a condition set which reduces the sample space. Firstly, let's set up the sample space and answer this without using the formal definition of conditional probability:

Of these four options, only three satisfy the condition of having at least one girl - this is the reduced sample space. Of these three options, only one results in the family having two girls. Therefore, the probability of the family having two girls is one third (1/3).

The conditional probability formula

In general, the conditional probability formula can be used to find the probability of event A occurring given event B has already occurred:

Multiplication rule of probability

The formula for conditional probability can also be rearranged to give the multiplication rule of probability:

Interesting results of conditional probability

The following observations are interesting results that will tell us about the relationship between two events in conditional probability:

• If Pr(A|B) = 1 then Pr(A ∩ B) = Pr(B); therefore, B ⊆ A. This means that when event B has occurred event A must have occurred because all elements in B are also contained in A.
• If Pr(A|B) = 0 then Pr(A ∩ B) = 0; therefore, events A and B are mutually exclusive. This means that when event B has occurred event event A cannot occur because no elements are common to both events (A ∩ B = ∅).
• If Pr(A|B) = Pr(A)/Pr(B) then Pr(A) = Pr(A ∩ B); therefore, A ⊆ B. This means that all elements contained within event A are also elements in event B.

13D - Exercises:

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