PROBABILITY
Probability is the chance that some event occurs. In statistics, probability is expressed by a number zero to one (0 being no chance, 1 being certain). Furthermore, finding the probability of something not happening is as simple as finding the compliment of that event, or taking the numerical probability of it happening and subtracting it from one. However, probability gets even more interesting when comparing two events. If they’re disjoint events, the probability of them happening together is zero. If they’re independent, the likelihood of one event does not affect the other.
When calculating the probability relationships between two events, it is important to keep in mind the probability rules of statistics. If you find yourself wondering about the “and”, or the probability that both events occur, you multiply the probabilities that they occur together. On the contrary, if you find yourself wondering about the “or”, or the probability that one events occurs but not the other, you add the probabilities that they occur together. Finally, if you want to know the “given”, or the probability that something happens given the other, you divide the probability of the event you’re looking for by the given event’s probability.
Probability is a broad and sometime complex topic, so it is common to encounter some confusion. Aside from looking for buzzer words such as “and” “or” and “given”, it is important to note how the probabilities were found. If the statistician used sample data from some type of study or experiment, was it sampled with or without replacement? This would affect whether probability stays the same each sample, or whether it increases as you remove units from your study.
A test is created that detects a very rare form of cancer. The test has a 99% detection rate. However, the probability that it detects you have the cancer but you do not, or the probability of a false positive, is 5%.
REAL LIFE EXAMPLE
Here is a contingency table to display the probability of every outcome:
The chance of testing positive given that you have the cancer is 99%
BUT, the probability that you have cancer given you test positive is only .4%
So...
It’s in the wording and the math! This cancer is extremely rare; only 1 in 5000 people theoretically have it. Even though the positive detection rate is 99%, think about the amount of people that would detect positive… a very small amount. Therefore, that false positive rate of 5% is out of that very small group, which is an even smaller group: .4% of people.