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Probabilities may be either marginal, joint or conditional. Understanding their differences and how to manipulate among them is key to success in understanding the foundations of statistics.
Marginal probability: the probability of an event occurring (p(A)), it may be thought of as an unconditional probability. It is not conditioned on another event. Example: the probability that a card drawn is red (p(red) = 0.5). Another example: the probability that a card drawn is a 4 (p(four)=1/13).
Joint probability: p(A and B). The probability of event A and event B occurring. It is the probability of the intersection of two or more events. The probability of the intersection of A and B may be written p(A ∩ B). Example: the probability that a card is a four and red =p(four and red) = 2/52=1/26. (There are two red fours in a deck of 52, the 4 of hearts and the 4 of diamonds).
Conditional probability: p(A|B) is the probability of event A occurring, given that event B occurs. Example: given that you drew a red card, what’s the probability that it’s a four (p(four|red))=2/26=1/13. So out of the 26 red cards (given a red card), there are two fours so 2/26=1/13.
How to Manipulate among Joint, Conditional and Marginal Probabilities
The equation below is a means to manipulate among joint, conditional and marginal probabilities. As you can see in the equation, the conditional probability of A given B is equal to the joint probability of A and B divided by the marginal of B. Let’s use our card example to illustrate. We know that the conditional probability of a four, given a red card equals 2/26 or 1/13. This should be equivalent to the joint probability of a red and four (2/52 or 1/26) divided by the marginal P(red) = 1/2. And low and behold, it works! As 1/13 = 1/26 divided by 1/2. For the diagnostic exam, you should be able to manipulate among joint, marginal and conditional probabilities.
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Bayes’ Theorem
Bayes’ theorem: an equation that allows us to manipulate conditional probabilities. For two events, A and B, Bayes’ theorem lets us to go from p(B|A) to p(A|B) if we know the
marginal probabilities of the outcomes of A and the probability of B, given the outcomes of A. Here is the equation for Bayes’ theorem for two events with two possible outcome (A and not A).
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Bayes’ Theorem Example
Let’s assume we know that 1% of women over the age of 40 have breast cancer.
[p(cancer)=0.01]
Let’s assume that 90% of women who have breast cancer will test
positive for breast cancer in a mammogram.
[p(positive test|cancer)=0.9]
Eight percent ofwomen that do NOT have cancer will also test positive.
[p(positive test|no cancer)=0.08]
What is the probability that a woman has cancer if she tests positive [p(cancer|positive test)]?
We will call p(cancer) = P(A), and the P(positive test) = P(B). We want to know P(A|B)–the probability of having cancer if you have a positive test.
Using Bayes’ theorem, we calculate that the likelihood that a woman has breast cancer, given a positive test equals approximately 0.10. This makes intuitive sense as (1) this result is greater than 1% (the percent of breast cancer in the general public).
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Photo credit: Matthew J. Keedy, Trinidad and Tobago.
