Conditional probabilities can be computed using the methods developed above if the appropriate information is available. Some times you will however have some information available, such as \(P(A | B)\) but need \(P(B | A)\text{.}\) The ability to "play around with history" by switching what has been presumed to occur leads to an important result known as Baye's Theorem.
Let \(S = \{ S_1, S_2, ... , S_m \}\) where the \(S_k\) are pairwise disjoint and \(S_1 \cup S_2 \cup ... \cup S_m = S\) (i.e. a partition of the space S). Then for any \(A \subset S\)
The conditional probability \(P(S_j | A)\) is called the posterior probability of \(S_k\text{.}\)
Notice, by the definition of conditional probability and the multiplication rule
But using the disjointness of the partition
Put these two expansions together to obtain the desired result.
To illustrate this result, from the web site http://stattrek.com/probability/bayes-theorem.aspx consider the following problem:
Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent years, it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain for tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time. When it doesn't rain, he incorrectly forecasts rain 10% of the time. What is the probability that it will rain on the day of Marie's wedding?
Notice, all days can be classified into one of two disjoint options:
In the notation of Bayes Theorem, let A represent a forecast of Rain and note you have
and
Further, you are given the conditional probabilities
Notice that the question provided requests that you find the probability of Rain given that the weatherman has forecasted rain. What is given on the other hand is the reverse of that conditional probability. Using Bayes Theorem allows you to turn this around...
Hence, putting these together gives
So, normally there is only a 5 percent chance of rain on a given day but given that the weatherman has forecast rain, the chance of rain has risen to a little more than 11 percent.
Let's try a Bayes Theorem example...
You have to be careful to extract the conditional probabilities from the problem.
Here is a more extensive Bayes Theorem example...
Notice that having the data expressed in tabular form sometimes makes it easier to deal with.
The interactive cell below can be used to easily compute all of the conditional probabilities associated with Bayes's Theorem. Notice how the relative size of the pie-shaped partition changes when you presume that an event in the space has already occurred.
Checkpoint 4.6.4 Insured vs Accident
Your automobile insurance company uses past history to determine how to set rates by measuring the number of accidents caused by clients in various age ranges. The following table summarizes the proportion of those insured and the corresponding probabilities by age range: One of your family friends insured by this company has an accident.
Age
Proportion of Insured
Probability of Accident
16-20
0.05
0.08
21-25
0.06
0.07
26-55
0.49
0.02
55-65
0.25
0.03
over 65
0.15
0.04
Plug the middle column into the first input box and the right column into the second input box of the Bayes Sage Cell
Checkpoint 4.6.6 Spinal bifida odds
Congratulations...your family is having a baby! As part of the prenatal care, some testing is part of the normal procedure including one for spinal bifida (which is a condition in which part of the spinal cord may be exposed.) Indeed, measurement of maternal serum AFP values is a standard tool used in obstetrical care to identify pregnancies that may have an increased risk for this disorder. You want to make plans for the new child's care and want to know how serious to take the test results. However, some times the test indicates that the child has the disorder when in actuality it does not (a false positive) and likewise may indicate that the child does not have the disorder when in fact it does (a false negative.)
The combined accuracy rate for the screen to detect the chromosomal abnormalities mentioned above is approximately 85% with a false positive rate of 5%. This means that (from americanpregnancy.org)