Conditional probability
- Given that means you are no longer looking at the whole sample space.
- You only look inside the condition, then count what also matches the target.
After the bar is the denominator
. The event after the bar is the new world.
- In , divide by .
- In , divide by .
- These are usually different. The order matters.
Worked example
First rung: shrink the denominator
A class has 12 boys and 18 girls. Of the girls, 6 study music. Find the probability that a chosen student studies music, given that the student is a girl.
- The words given that the student is a girl shrink the denominator to the 18 girls.
- Inside that smaller group, 6 study music.
Answer
Worked example
Read the condition from the table
A student is chosen from a music table. Find the probability that the student is male, given that the student plays drums. There are 11 male drummers and 31 drummers in total. (9709/52 O/N 2021 Q1)
- The condition is plays drums, so the denominator is the drums total.
- Of the 31 drummers, 11 are male.
Answer
| Data type | How to find the conditional probability |
|---|---|
| Two-way table | target-and-condition cell over condition total |
| Venn diagram | overlap region over the whole condition circle |
| Possibility grid | winning cells inside the condition cells |
| Tree diagram | target path(s) over all condition path(s) |
independent: P(A∩B)=P(A)P(B)
P(A│B) = P(A∩B)/P(B) = 0.20/0.40 = 0.500
Common mistake
The most common error is using the grand total as the denominator. Once a question says given that B, the denominator is B, not everyone.
Why this also explains independence
- If , then B did not change the chance of A.
- That is exactly what independent means.
- If the conditional probability changes, the events are dependent.