The probability tree and counting board lets you fill a bag with labelled counters, choose how many you draw, and see every path with its probability written as an exact fraction. It also adds up the paths so you can check the total.
Use it to see why a second branch changes when a counter is not put back.
How do you use it?
- In Bag contents, write label:count pairs separated by commas, for example Red:3, Blue:2. Use up to 5 labels and whole-number counts.
- Choose the number of draws: 1, 2 or 3.
- Choose After each draw: do not replace the counter, or replace it.
- Under Event, type the label you care about and how many times it should appear, for example Red and 2.
- Press Build the tree. Press Reset to return to the sample bag.
How do you read the result?
The first line gives the probability of your event, such as P(exactly 2 × Red) = 3/10, and lists the paths that belong to it.
The probability tree shows each branch with its fraction. The outcome table lists every path, the multiplication that gives it, and its probability. Paths that belong to your event are shaded.
Below the table, the total-probability check adds every path. It should give 1. When the draws are without replacement, a conditional change note points out how the second fraction differs from the first.
Example walk-through
The sample is 3 red and 2 blue counters, two draws, not replaced, event “exactly 2 Red”.
| Path | Working | Probability |
|---|---|---|
| Red → Red | 3/5 × 1/2 | 3/10 |
| Red → Blue | 3/5 × 1/2 | 3/10 |
| Blue → Red | 2/5 × 3/4 | 3/10 |
| Blue → Blue | 2/5 × 1/4 | 1/10 |
The tool reduces 2/4 to 1/2 as it goes. Only the first path has two reds, so P(exactly 2 Red) = 3/10. The four paths add to 3/10 + 3/10 + 3/10 + 1/10 = 1.
The conditional note explains the key idea: after a red is taken, the bag holds 4 counters with 2 red, so the next chance of red is 1/2, not 3/5.
Now change the last choice to replace the counter. The bag is restored each time, so every branch repeats 3/5 and 2/5.
The four paths become 9/25, 6/25, 6/25 and 4/25, which also total 1. Then set the event to “exactly 1 Red”: the two mixed paths give 6/25 + 6/25 = 12/25. Without replacement the same event gives 3/10 + 3/10 = 3/5, so the choice changes the answer.
What are the assumptions and limits?
- Every counter is equally likely to be drawn. The tool models an ideal bag.
- Counts must be zero or positive whole numbers. Negative or fractional counts are rejected with a message.
- Labels must be different, and you can use at most 5.
- Without replacement you cannot draw more counters than the bag holds.
- Inheritance grids are models, not personal genetic advice.
Which lessons explain the ideas behind it?
- Represent mutually exclusive outcomes sets up what a path and an outcome are.
- Use a two-stage probability tree builds the tree by hand.
- Explain when multiplication needs independence explains why the with-replacement tree multiplies the same fractions.
- Update probabilities after an item is not replaced explains the conditional change the tool points out.
The wider topic is probability reasoning, and the mixed practice set lets you test yourself. Biology students meeting inheritance grids can start from the Biology learning guide.
If you want a teacher to check how you set up your own trees, see online one-to-one Mathematics tuition. Other tools are in the learning tools directory.