P(A ∩ B) = P(A)P(B) is true at exactly one overlap and false everywhere else. Here is the general formula that is always true.
One hundred students. 40 take Higher Level and 50 play a sport, and those two totals never change. Slide the overlap and compare the two readouts.
The two totals never move. Only how much they overlap.
P(A | B) = P(A ∩ B)P(B), which rearranges to the form that is usually more useful:
P(A ∩ B) = P(B) P(A | B). This one is always true. It needs no assumption at all.
P(A ∩ B) = P(A)P(B) is the special case. It holds only when A and B are independent, which is the same as saying P(A | B) = P(A): knowing B tells you nothing about A. Independence is something you check or are told, never something you assume because multiplying is easier.
Two readings of the same bar. P(A | B) and P(B | A) are different numbers here, 0.5 and 0.625, because they divide by different things. Swapping them is the error behind almost every misread statistic in the news.
1. With the overlap at 20 students, find P(A | B).
2. What overlap makes A and B independent? Give the number of students.
3. P(A) = 0.6, P(B) = 0.3 and P(A ∩ B) = 0.18. The events are:
Using P(A ∩ B) = P(B)P(A | B) rather than multiplying, unless independence has been established. Writing the general form costs nothing and is always right.
Testing independence properly: compare P(A | B) with P(A), or P(A ∩ B) with P(A)P(B). Either earns it; a vague appeal to the context does not.
Keeping P(A | B) and P(B | A) apart, and dividing by the probability of the thing you were told.
These pages are free and stay free, but they are general and your IA is not. Send me your research question, or whatever exists so far, and I will tell you in writing whether the topic has a ceiling on it, where the marks are going, and what to change first. That costs nothing and it comes back within 24 hours.
Written by a serving IB Diploma and Career-related Programme Coordinator and Head of Mathematics, who reads internal assessments across every subject group every year. If you then want the whole draft reviewed properly against all five criteria, that is the paid one, and it is refunded if it does not name at least three specific things to fix.
Get a free verdict Full written review, $99
I never write any part of it. Not a sentence, not a calculation, not your data. Under 18: a parent buys this and the thread is with them. I do not work with students at my own school.