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Probability Tree Explorer

Run the Probability Tree Explorer Fullscreen Edit the Probability Tree Explorer in the p5.js Editor

Description

This interactive MicroSim helps students understand compound probability by visualizing a probability tree for drawing two balls from a bag without replacement. The key insight is that branch widths narrow with each level, making the multiplication of probabilities visually intuitive.

Pedagogical Intent

Students often struggle to understand why we multiply probabilities along branches in a probability tree. This MicroSim addresses that challenge by:

  1. Visual Width Representation: Branch widths are proportional to probability, so students can see how the "probability flow" narrows
  2. Dynamic Conditional Probabilities: Changing the ball counts shows how removing a ball affects the second draw
  3. Calculation Transparency: Hovering over outcomes reveals the step-by-step multiplication

Features

  • Interactive Sliders: Adjust the number of red (1-10) and blue (1-10) balls
  • Probability Tree Visualization: Shows all possible outcomes (RR, RB, BR, BB)
  • Proportional Branch Widths: Wider branches = higher probability
  • Path Highlighting: Hover over outcome nodes to highlight the path
  • Calculation Panel: See the step-by-step probability calculation for each path
  • Show All Calculations: Toggle to display all four outcome probabilities at once

Scenario

A bag contains red and blue balls. You draw two balls one at a time without replacement (the first ball is not returned before drawing the second). The tree shows:

  1. First Level: Probability of drawing each color first
  2. Second Level: Conditional probability of each color given the first draw
  3. Outcome Nodes: Final compound probabilities for all four possible outcomes

How to Use

  1. Adjust the sliders to set the number of red and blue balls
  2. Observe the tree - notice how branch widths change with different ball counts
  3. Hover over outcome nodes (RR, RB, BR, BB) to see the calculation
  4. Check "Show Calculations" to see all probabilities simultaneously
  5. Click "Reset" to return to default values (5 red, 3 blue)

Embedding This MicroSim

You can include this MicroSim on your website using the following iframe:

1
<iframe src="https://dmccreary.github.io/automating-instructional-design/sims/probability-tree-explorer/main.html" height="570px" scrolling="no"></iframe>

Lesson Plan

Learning Objective

Students will be able to calculate compound probabilities by tracing paths through a probability tree and multiplying branch probabilities.

Bloom's Taxonomy Level

Analyze (Level 4) - Students break down compound probability problems into sequential events and determine the relationship between conditional and compound probabilities.

Prerequisites

  • Understanding of basic probability (favorable outcomes / total outcomes)
  • Familiarity with fractions and decimal conversion

Activities

1. Exploration (5 minutes)

Have students explore the default configuration (5 red, 3 blue balls):

  • What is the probability of drawing red first?
  • Why is P(RR) different from P(RB)?
  • What do you notice about the branch widths?

2. Investigation (10 minutes)

Students answer guided questions:

  • Set red = 5, blue = 5. What patterns do you notice?
  • What happens when red = 1? What is P(RR)?
  • When does P(RR) = P(BB)?

3. Calculation Practice (10 minutes)

Without hovering, have students calculate:

  • P(BR) when there are 4 red and 6 blue balls
  • Verify their answer by hovering over the BR node

4. Discussion (5 minutes)

  • Why do we multiply probabilities along a path?
  • Why is "without replacement" different from "with replacement"?
  • Why do all four outcome probabilities sum to 1?

Assessment Questions

  1. If there are 3 red and 2 blue balls, what is P(RB)?
  2. Explain why P(Red on second draw | Red on first draw) is different from P(Red on first draw)
  3. For what combination of balls is P(RR) = P(BB)?

Extensions

  • Compare to "with replacement" (probabilities stay constant)
  • Extend to three draws - what would the tree look like?
  • Calculate P(at least one red) using the complement rule

Mathematical Background

For drawing without replacement from a bag with R red and B blue balls (total T = R + B):

First Draw Probabilities:

  • P(Red first) = R/T
  • P(Blue first) = B/T

Conditional Probabilities (Second Draw):

  • P(Red second | Red first) = (R-1)/(T-1)
  • P(Blue second | Red first) = B/(T-1)
  • P(Red second | Blue first) = R/(T-1)
  • P(Blue second | Blue first) = (B-1)/(T-1)

Compound Probabilities:

  • P(RR) = (R/T) x ((R-1)/(T-1))
  • P(RB) = (R/T) x (B/(T-1))
  • P(BR) = (B/T) x (R/(T-1))
  • P(BB) = (B/T) x ((B-1)/(T-1))

Note that P(RR) + P(RB) + P(BR) + P(BB) = 1 (certainty).

References