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STAT110Probability and Statistics
ExampleBayes' theorem
How to work out how likely something is once you have seen new evidence.
Picture 10,000 people screened for an illness that 1 in 100 has, with a test that is right 99% of the time. Of the 100 ill people 99 test positive, and so do 99 of the 9,900 healthy people, by mistake.
So a positive result puts you among 198 people, of whom only 99 are ill: a 50% chance.
The 10,000 people split by illness, then by test result; the highlighted groups are everyone who tests positive, 99 ill and 99 healthy.
How to spot it
The question gives the chance of a result for each possible cause (a positive test if ill, a positive test if healthy) and asks for the chance of the cause once you have seen the result.
Worked example
The same test (positive for 99% of ill people, for 1% of healthy people) is used on 10,000 people, but now 2% of them have the illness. What is the probability that a person who tests positive is ill?
Split the population into the possible causes using how common each is.
Ill: of . Healthy: .
For hypotheses that partition the sample space and data with :
The denominator is the law of total probability. is the prior, the likelihood, and the posterior.
In general .
On the 10,000 people: .
Same test: it flags 99% of ill people and wrongly flags 1% of healthy people. Now you choose how common the illness is among the 10,000.
Move the slider until a positive test is a coin flip: a 50% chance of being ill.
- how common it is (%)
- 0.1
- chance you're ill (%)
- 9.0
At 1%, the 99 ill people who test positive exactly match the 99 healthy people flagged by mistake. Any rarer, and most positive tests are false alarms.