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 that says
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.
The
line 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.
Let's Practice:
- 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 |
0.30 |
0.13 |
0.01 |
| Child not in College |
0.20 |
0.25 |
0.11 |
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 as 0
.30
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 of 4/51.
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)
P(ace
queen) =
P(ace
queen) is found using the multiplication rule. Since the problem states that there is no replacement we can find
P(ace
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.