Multi-stage experiments and path rules

What is it about?

A multi-stage experiment repeats a simple one, e.g. drawing twice from an urn with $3$ red and $2$ blue balls. The tree diagram shows every possible course; each path is one outcome.

The branches carry probabilities. With replacement they stay the same; without replacement the urn shrinks (after red: $\frac24$ red, $\frac24$ blue).

• 1st path rule (product): without replacement $P(rr) = \frac35 \cdot \frac24 = \frac{3}{10}$, with replacement $\frac35 \cdot \frac35 = \frac{9}{25}$.
• 2nd path rule (sum): $P(\text{same colour}) = \frac{3}{10} + \frac{1}{10} = \frac25$ (without replacement).
• Complement: $P(\text{at least one red}) = 1 - P(bb) = \frac{9}{10}$.
• Special cases without replacement: a used-up colour gets a branch with probability $0$ (dashed), and the urn runs empty.

In the simulation the relative frequency approaches the probability; the funnel shows the $95\,\%$ range.

Try it

1. Draw once and watch the path light up. Draw
2. Compare both trees side by side. Comparison
3. Play the calculation for “same colour”. Path rules
4. Draw 1000 times. Simulation

Tasks

Task 1: Path probabilities

$3$ red, $2$ blue, two draws without replacement: compute all four path probabilities. Load task

Task 2: At least one red

Three draws with replacement: find $P(\text{at least one red})$. Load task

Ideas for the classroom

• Drawing a few times from a real urn first and then letting the simulation do the many repetitions can work well.
• Common stumbling blocks: keeping the same probabilities without replacement; adding along a path instead of multiplying; treating $rb$ and $br$ as one outcome.