Trees with conditions.

Step 1 of 5

Splitting by illness

In a sports hall $1000$⁠ people stand and take a test for a disease. The chart beside the Scene is a tree of the situation: from the root, $1000$⁠, two branches go out, ill and healthy, and from each of those two more, a positive and a negative result. Instead of chances, the branches carry numbers of people, which makes the sums much easier.

Here How common the disease is is $10\%$⁠, and Test accuracy $90\%$⁠: the test catches $90\%$⁠ of the ill and clears $90\%$⁠ of the healthy. The ill are shown in red.

The rate on every second branch is a conditional chance: $90\%$⁠ is the chance of a positive result given that the person is ill.

Step 2 of 5

The other branch

The healthy take the test too, and it gets $10\%$⁠ of them wrong. Someone healthy who tests positive is a false positive. Someone ill who tests negative is a false negative.

Step 3 of 5

Every positive

The board now walks every positive out to the two mats at the front: the ill on the left, the healthy on the right. Two paths on the tree end in a positive result, and the number of positives is the sum of both.

Step 4 of 5

Without putting back

The same trees work for any experiment in stages. On every branch write the chance given everything that happened earlier on the path, and multiply along a path:

$P(A \cap B) = P(A) \cdot P(B \mid A)$$P(A \cap B) =$$P(A) \cdot P(B \mid A)$⁠

This is the multiplication rule in full. When the stages are independent, $P(B \mid A) = P(B)$⁠, and it goes back to the simple rule. When you draw marbles from a bag without putting them back, the second stage depends on the first.

Step 5 of 5

Along a path

Back to the hall, this time with a disease that affects $5\%$⁠. The path “ill, then positive” is two branches in a row, so multiply the two rates.