Loading...
Loading...
Master the concept of conditional probability - finding probabilities when you have additional information.
Conditional probability answers the question: "Given that we know something happened, how does that change the probability of something else?"
Read as "the probability of B given A"
P(B|A) = Probability of B occurring, given that A has occurred
P(A∩B) = Probability that BOTH A and B occur
P(A) = Probability that A occurs (must be greater than 0)
Think of conditional probability as "zooming in" on the sample space. When we know A happened, we ignore all outcomes where A didn't happen and focus only on outcomes where A occurred.
P(Disease | Positive Test) - What's the chance you have a disease if you test positive? This is crucial for understanding test results.
P(Rain | Cloudy) - Given it's cloudy, what's the probability of rain? Forecasters use conditional probabilities constantly.
P(Defect | Monday Production) - Are items made on Monday more likely to be defective? Helps identify production issues.
P(Spam | Contains "FREE") - If an email contains "FREE", is it likely spam? This powers modern spam filters.
You draw a card from a standard deck and are told it's a face card (J, Q, or K). What is the probability that it's a King?
The probability of drawing a King given that it's a face card is 1/3 or about 33.33%.
These are NOT the same! This is called the "fallacy of the converse."
Example:
When we condition on A, we must only consider outcomes where A occurred.
Example: When told a die roll is even (2, 4, or 6), don't use 6 total outcomes - use only the 3 even outcomes!
P(A∩B) is the probability of BOTH occurring. P(B|A) is the probability of B when we KNOW A occurred.
If P(B|A) = P(B), then A and B are independent - knowing A doesn't change B's probability.
Bayes' theorem lets you "reverse" conditional probabilities:
Rearranging the conditional formula:
Useful when A has multiple cases: