Independence as something you check, not something you assume because multiplying is easier.
Ask them to find the overlap where multiplying is allowed.
Let them hunt. The two bars match at exactly one position, 20 students, and nowhere else. That is independence discovered rather than announced, and it is a much better definition than the one in the formula book.
The totals, 40 and 50, never change throughout. Only the overlap does, which makes the point that independence is about the relationship between the events and not about their sizes.
P(A ∩ B) = P(B) P(A | B) needs no assumption whatsoever. P(A)P(B) is the special case that holds only under independence.
Get them writing the general form by default. It costs one extra symbol and it is never wrong, whereas the habit of multiplying is wrong most of the time and feels right every time.
| 1. P(A | B) with overlap 20 | 20/50 = 0.4, which here also equals P(A), so that is the independent case. |
| 2. The independent overlap | P(A)P(B) = 0.4 × 0.5 = 0.2, so 20 students. |
| 3. P(A)=0.6, P(B)=0.3, P(A∩B)=0.18 | A, independent. 0.6 × 0.3 = 0.18 exactly. |
1 markUsing P(A ∩ B) = P(B)P(A | B) rather than multiplying, unless independence is established.
1 markTesting independence properly, by comparing P(A | B) with P(A) or P(A ∩ B) with P(A)P(B).
1 markDividing by the probability of the event you were told, not by the whole sample space.
| They give | What it means |
|---|---|
| 0.5 for P(A | B) | They computed P(B | A). The bar divides by what you were TOLD. Make them say the sentence "given B, B is now everything" out loud. |
| 0.2 for P(A | B) | Divided by 100 rather than by 50, so they found the intersection, not the conditional. |
| "Mutually exclusive" (Q3) | Mutually exclusive needs P(A ∩ B) = 0. Worth separating firmly, because the two words get stored in the same place. |
| "Dependent" (Q3) | They did not run the test. 0.6 × 0.3 is 0.18, which matches exactly. |
| 0.2 for Q2 | Right probability, wrong unit. The question asked for students. |
"How do I know if they're independent?" You check, or you are told. There is no third option, and a question that wants you to assume it will say so.
"Is P(A|B) the same as P(B|A)?" No, and this is worth two minutes of real-world examples. P(ill | positive) and P(positive | ill) differ enormously, which is the whole of 4.13 waiting.
"Does mutually exclusive mean independent?" The opposite, in fact: if they are mutually exclusive then knowing B tells you A definitely did not happen, which is maximum dependence.
| What is happening | |
|---|---|
| 1 | Hunt for the overlap where multiplying works. Let them find 20. |
| 2 | The two formulae, with the general one written first on the board. |
| 3 | The three questions, with P(A|B) and P(B|A) computed side by side every time. |
| 4 | Past paper questions where independence must be tested rather than assumed. |
| 5 | Set up Bayes: if P(A|B) and P(B|A) differ, there must be a rule connecting them. |
Do not write P(A ∩ B) = P(A)P(B) on the board without the independence condition attached. It will be copied without it.
Do not let "independent" and "mutually exclusive" be used interchangeably in discussion, even casually.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.