Independent Events and Conditional Probability Notation

Checking what a diagram suggests

You already know how to read a two-way table, a tree and a Venn diagram, and how to reason with them without any symbols at all. That was deliberate, because it means you now know what the combinations mean before meeting the shorthand for them. Here is what the shorthand adds. A diagram can make two things look unrelated, and looking unrelated is an impression rather than a finding. The notation turns it into a test: multiply the two probabilities and see whether you get the probability of both happening together. If you do, the events are independent, and you have established something rather than sensed it. The same is true of conditional probability. Saying that a group behaves differently is an impression; writing the probability of one thing given another, and calculating it, is a claim you can defend. So the symbols are not an extra layer of difficulty laid over the diagrams. They are how you check what the diagram suggested, and they let you combine probabilities once you no longer need to draw anything.

The notation you need

Five pieces of notation and the two ideas behind them. The vertical bar is the one worth slowing down for.

Match the notation to the words

What the bar actually asks

A question asks for P(B|A). What is it asking you to do?

• Look only at the cases where A happened, and find what fraction of those also had B
• Find the probability that A and B both happen
• Find the probability that either A or B happens
• Find the probability of A given B

Independent, or only unrelated-looking

Two hypothetical two-way tables. Both look ordinary. Only one describes independent events, and the only way to know which is to do the test.

Which two show independence

Select the TWO things that would establish that events A and B are independent.

• P(A) x P(B) works out to exactly P(A and B)
• P(B|A) works out to exactly the same as P(B)
• The two events seem unconnected in real life
• The numbers in the table are spread fairly evenly

The order that changes the answer

Two things about this notation are worth fixing in your mind before you use it. First, the order inside the bracket matters, and it is the commonest place marks are lost. P(B|A) and P(A|B) are different questions and usually have different answers. Take the clubs table above. P(X|A) asks: of the students in club A, what fraction are also in X? There are 40 in A and 30 of those are in X, so the answer is 30 out of 40, which is 3/4. Now P(A|X) asks something else entirely: of the students in X, what fraction are in A? There are 50 in X and 30 of those are in A, so the answer is 30 out of 50, which is 3/5. Same cell on the top, different denominators, different answers. Read the question twice and identify which event you are being given before you write anything down. Second, a note on tiers. Your specification marks this formal notation as Higher-tier content, but it appears on papers at both tiers, and it says so explicitly. So if you are working at Foundation, do not assume this material is somebody else’s problem: you can be asked about it, and the reasoning behind it is the same reasoning you already used on the diagrams.

The two formulas in words

For independent events, P(A and B) equals P(A) _____ P(B). That is a _____ rather than an assumption: work out both sides and see whether they match. For conditional probability, P(B|A) equals P(A and B) _____ P(A), and the vertical bar means _____. Because the bar restricts you to the As, the _____ changes, which is why the answer differs from P(B).

Which conditional is this

Using the clubs table (30 in both A and X, 10 in A only, 20 in X only, 40 in neither), what is P(X|A)?

• 3/4
• 3/5
• 3/10
• 2/5

The sentence that swaps them

Four sentences about the clubs table. Select the ONE that states a conditional the wrong way round.

• Of the 40 students in A, 30 are also in X, so P(X|A) is 3/4.
• Of the 50 students in X, 30 are also in A, so P(A|X) is 3/5.
• Since 30 of the 40 students in A are also in X, P(A|X) is 3/4.
• P(A and X) is 30 out of all 100 students, which is 3/10.

Testing a table you already know

Here is the whole method applied to data you have already met. The travel and lunch table recorded 200 students: 48 came by bus and brought a packed lunch, 32 came by bus and used the canteen, 72 walked and brought a packed lunch, and 48 walked and used the canteen. When you first met it you read probabilities off it. Now test something you could not before. Are travelling by bus and bringing a packed lunch independent? First find the pieces. There are 80 bus users out of 200, so P(bus) is 2/5. There are 120 packed lunches out of 200, so P(packed) is 3/5. The cell for both is 48 out of 200, which is 6/25. Now apply the test: 2/5 multiplied by 3/5 is 6/25, which is exactly what the cell gives. So the events are independent, and you can say so rather than guessing. Check it the other way as confirmation. P(packed|bus) restricts you to the 80 bus users, of whom 48 bring a packed lunch, giving 48 out of 80, which is 3/5 - precisely P(packed). Knowing that a student came by bus tells you nothing you did not already know about their lunch. That is what independence means, and notice that the table alone never said so: two sets of numbers looked plausible, and only the calculation settled it.

State the test for independence

Assemble a sentence stating what independence means and how it is established.

Working out which formula

A question gives you a two-way table and asks several things about it. Take the decisions in order.

• It asks for the probability that a randomly chosen person is in both categories. What do you use?
• It then says "given that the person is in A". What changes?
• It asks whether the two events are independent. How do you answer?
• Your answer to a conditional looks surprisingly large. What is the first thing to check?

Explain the notation

A friend can read a two-way table but freezes whenever the symbols appear. Explain what the notation means and what it lets them do that a diagram alone does not.

• Explain what P(A), P(A and B) and P(B|A) each mean in words
• Explain what the vertical bar does to the group you are looking at, and therefore to the denominator
• State the test for whether two events are independent
• Explain why that is a test rather than an impression from the table
• Give a second way of expressing independence, using a conditional probability
• Explain why P(B|A) and P(A|B) usually differ, with an example of your own
• Note that this notation is Higher-tier content but appears on papers at both tiers
• Finish by saying what the notation lets you do that a diagram alone does not