Conditional Probability
- Probability values change when information about a related event is known.
- Recalculate using the updated conditions or remaining outcomes (examples: sequential draws, medical tests).
- Context and prior information affect how likely an event is.
Definition
Section titled “Definition”Conditional probability is the probability of an event occurring given that another event has already occurred.
Explanation
Section titled “Explanation”When an event is known to have happened, the sample space or the relevant counts used to compute probabilities can change, and the probability of a subsequent event should be computed under that updated condition.
Example illustrations in the source:
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For a bag with 10 red balls and 5 blue balls, the probability of picking a red ball without any prior information is 10/15 or 2/3. If the first ball picked from the bag was blue, the probability of picking a red ball on the second pick changes because the known outcome affects the remaining composition. According to the example, there are now only 4 balls left in the bag, and only 4 of them are red, so the probability of picking a red ball on the second pick, given that the first ball was blue, is 4/9.
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In medical testing, a disease that occurs in only 1% of the population and a test that is 99% accurate illustrate how prior likelihoods matter. The probability of a positive test result depends on the person’s overall likelihood of having the disease: with no symptoms and no family history the probability of having the disease is very low (so the probability of a positive test result, given that they do not have the disease, is also very low), whereas with symptoms and a family history the probability of having the disease is much higher (so the probability of a positive test result, given that they do have the disease, is also much higher).
Examples
Section titled “Examples”Drawing balls from a bag
Section titled “Drawing balls from a bag”- Bag contents: 10 red balls and 5 blue balls.
- Probability of picking a red ball without prior information: 10/15 or 2/3.
- Given the first ball picked was blue, the example states there are now only 4 balls left and only 4 of them are red, yielding a conditional probability of 4/9 for picking a red ball on the second pick.
Medical testing
Section titled “Medical testing”- Disease prevalence: 1% of the population.
- Test accuracy: 99% accurate.
- The probability of a positive test result depends on prior likelihood: with no symptoms and no family history the chance of having the disease (and therefore a positive result given disease) is very low; with symptoms and family history the chance is much higher (and so is the probability of a positive result given disease).
Notes or pitfalls
Section titled “Notes or pitfalls”- It is important to consider the context and all available information when calculating the probability of an event.