Why does the probability of A given B not necessarily equal the probability of B given A?
Conditions
- A and B are events in a probability space.
- and .
Reasoning, step by step
- Identify that measures the likelihood of A occurring when B is known to have occurred.
- Identify that measures the likelihood of B occurring when A is known to have occurred.
- Recognize that these two quantities use different denominators ( vs ) in their definitions.
- Apply Bayes' theorem to link them: .
- Conclude that unless , the two conditional probabilities are not equal.
Example
The script states: 'The probability of A given B need not equal the probability of B given A. Bayes’ theorem connects them with the prior event probabilities.'
Common misconceptions
- Assuming conditional probability is symmetric.
- Thinking that and are always numerically equal.
- Forgetting that the prior probabilities and act as scaling factors between the two conditionals.
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