Probability tree: the 3 rules to compute a probability without mistakes

The 3 rules

Rule 1 (along a branch): you multiply probabilities. $P(A \cap B) = P(A) \times P(B|A)$

Rule 2 (at a node): the probabilities leaving a node sum to 1.

Rule 3 (several branches → same event): you add the probabilities.

Step-by-step example

Urn: 3 red balls, 2 black. Draw 2 balls, without replacement. Probability of drawing 2 reds?

Tree:

• 1st draw: P(R) = 3/5, P(B) = 2/5.
• 2nd draw, if 1st = R: 2R + 2B left → P(R|R) = 2/4.
• 2nd draw, if 1st = B: 3R + 1B left → P(R|B) = 3/4.

Compute — RR branch: $P(RR) = \dfrac{3}{5} \times \dfrac{2}{4} = \dfrac{3}{10}$

Classic application

“One red and one black, any order” = sum of 2 branches: R→B and B→R.

• $P(RB) = 3/5 \times 2/4 = 6/20$
• $P(BR) = 2/5 \times 3/4 = 6/20$

Total: $12/20 = 3/5$.

Common mistakes

1. Adding along a branch instead of multiplying.
2. Forgetting to sum branches that lead to the same outcome.
3. Forgetting “without replacement”: 2nd draw depends on 1st.
4. Siblings not summing to 1: sign of an error.

?À toi de jouer

Bag: 4 white, 6 black. Draw 2 with replacement. P(2 whites)?

A.16/100 = 4/25B.8/10 = 4/5C.4/10 × 3/9 = 2/15D.16/90

Valider