Conditional probability

What conditional probability means Extended

A conditional probability is the probability of an event given that something else is already known to have happened.

Compare these two questions about a class:

• "A student is chosen at random. What is the probability that the student wears glasses?"
• "A student is chosen at random. Given that the student is a girl, what is the probability that she wears glasses?"

In the second, you are told the student is a girl, so the boys can be ignored completely. The question has become: what fraction of the girls wear glasses?

That is the whole idea. The given information shrinks the group you choose from. The bottom of the fraction is no longer everybody; it is only the members of the group you have been told about.

$$\text{probability}=\frac{\text{number in the given group who also have the property}}{\text{number in the given group}}$$

Questions use words such as "given that", "if", or "a student who ... is chosen". You do not need any special notation or formula for this topic: understanding which group you are now choosing from is enough.

Conditional probability from a table Extended

A two-way table sorts the same people in two ways at once, with totals for every row and column. To find a conditional probability:

1. Find the row or column for the given group, and read its total. That is the bottom of the fraction.
2. In that same row or column, read the number who also have the property asked about. That is the top.
3. Divide.

The most common mistake is to divide by the grand total, as if nothing had been given. The second most common is to divide by the wrong row or column: "given that the student is in Year 10" means divide by the Year 10 total, not by the football total.

Conditional probability from a Venn diagram Extended

On a Venn diagram the given group is usually a circle, or the part outside a circle.

• "Given that the person is in $B$": the bottom is the total of circle $B$ (both of its regions).
• "Given that the person is not in $B$": the bottom is everything outside $B$, including the region outside both circles.
• "Given that the person is in at least one set": the bottom is everything inside the circles.

Then the top is the part of that group that has the property asked about.

The method is identical when the regions hold probabilities instead of numbers. Probabilities are just the numbers divided by the total, so the ratio is the same: divide the probability of the part you want by the probability of the whole given group. For example, if $\mathrm{P}(B)=0.4$ and the overlap is $0.1$, then given $B$, the probability of $A$ is $\frac{0.1}{0.4}=0.25$.

Conditional probability on a tree diagram Extended

On a tree diagram for events that depend on each other, the second set of branches already shows conditional probabilities. In a bag of 3 red and 5 white counters taken without replacement, the branch "red" after a first red counter is $\frac{2}{7}$: that is the probability the second counter is red given that the first was red.

Harder questions turn this round and give you the result, asking about what came first: "given that the second counter is red, find the probability that the first was red." Use the same idea as before, with probabilities:

1. Find the probability of the given result by adding every path that leads to it. This is the given group.
2. Find the probability of the path or paths that also have the property asked about.
3. Divide the second by the first.

Why this works. Imagine the experiment done many times. The given result happens on a certain fraction of occasions; you only look at those. Among them, the fraction that also had the first property is the ratio of the two probabilities, exactly as dividing two counts in a table.