When dealing with two events (usually called A and B), sometimes the events are so related to each other, that the probability
of one depends on whether the other event has occurred. When we talk about probabilities based on the fact that something else has already happened we call this conditional probability
What changes when dealing with conditional probability
is that we know for certain that something else has already happened. This means that in our definition of probability
where the “total number of ways” is based on the fact that we know something else has already occurred.
There are two ways to approach conditional probability
and depends on the type of problem that you are given. In a situation where you are given percentages and probabilities (usually but not always in a table
format) we make use of the conditional probability
rule (shown below). In situations where you are trying to compute probabilities on your own (instead of them being given to you) most of the time it is easier to not to use the formula.
Conditional Probability Rule:
Consider events A and B.
between A and B is read “given”. So translated, this reads, "the probability
of A given that B has happened." The event on the right side
of the line
is the event that has already happened.
What The Rule Means:
This rule is applied when you have two events and you already know the outcome of one of the events. In doing the computations, you will need to be able to find the probability
of A and B, that is, P(A
B). Problems of this type make use of the multiplication rule. If you need help with the multiplication rule or understanding what type of problems make use of the rule, review the lesson on the Multiplication Rule
Let’s look at several examples that do and do not make use of the rule.
- A survey of 500 adults asked about college expenses. The survey asked questions about whether or not the person had a child in college and about the cost of attending college. Results are shown in the table below.
||Cost Too Much
||Cost Just Right
||Cost Too Low
|Child in College
|Child not in College
Suppose one person is chosen at random. Given that the person has a child in college, what is the probability
that he or she ranks the cost of attending college as “cost too much”?
This problem reads:
P(cost too much
child in college) or
P(cost too much given that there is a child in college)
P(cost too much child in college) =
P(cost too much
child in college) can be found from the table
P(child in college) can be found by adding 0.30 + 0.13 + 0.01 = 0.44
In the previous example notice that the denominator of the probability fraction
dealt with having a child in college. This was our given information. In other words, we already KNOW that the person has a child in college. So we aren’t talking about all 500 people in the sample, we’re only talking about the ones that have a child in college. In this problem we don’t know the actual numbers, but we do know percentages.
Now let’s look at an example where probabilities are not already given and we have to come up with our own as we work through the problem.
- Suppose you draw two cards from a standard deck without replacement. Given that the first card is an ace, what is the probability that the second card is a queen?
Let’s take a look at this problem without making use of the formula. Consider that we KNOW that an ace has already been pulled from the deck. This means there are now 3 aces in the deck of 51 cards that are left. Now consider the probability
of drawing a queen from that deck of 51. There are still four queens in the remaining deck of 51. This gives the probability
The problem could be solved using the conditional probability
rule as shown below. However, look at how much work is needed to use the formula rather than simply thinking through what the problem says.
This problems reads
P(ace given queen) or P(ace
queen) is found using the multiplication rule. Since the problem states that there is no replacement we can find
queen) as (4/52)(4/51) = 4/663
We know that
P(queen) = 4/52
So how will you make the determination of how to solve a conditional probability
problem? As a general rule, problems where information is given in a table
are best solved by using the formula. Problems where you have to figure out probabilities as you work through the problem are best solved by adjusting for the information you already know.