Dependent probability visualisation tool

Add or delete balls from the canvas. Drag them into or out of the box.

Explore probability by changing the contents of the box

Probability is easier to understand when you can see what is happening.

This probability visualisation tool lets you add red, green and blue balls, then drag them into or out of the box. As the contents of the box change, you can compare the number of each colour and think about how likely each outcome would be if one ball were chosen at random.

For example, suppose the box contains:

• 4 red balls
• 3 green balls
• 1 blue ball

There are 8 balls altogether. The probability of choosing a red ball is therefore 4/84/84/8, the probability of choosing a green ball is 3/83/83/8, and the probability of choosing a blue ball is 1/81/81/8.

Try changing the number of balls. Which colour is most likely to be selected? Can you make two colours equally likely? Can you create an event with a probability of exactly 1/21/21/2?

Using the visualiser to understand probability

The probability of an event can be written as:

number of successful outcomes ÷ total number of possible outcomes

If every ball in the box is equally likely to be chosen, the probability of selecting a particular colour depends on how many balls of that colour are present.

Adding another red ball increases the probability of choosing red. Removing a green ball reduces the probability of choosing green.

The important point is that the probability changes when the contents of the box change.

The visualiser allows you to experiment with this idea rather than simply reading probabilities from a question.

Try these probability experiments

Use the tool to investigate the following questions.

Make an event with probability one half

Can you arrange the balls so that the probability of choosing a red ball is exactly 1/21/21/2?

There is more than one possible answer. For example, 3 red balls out of 6 balls altogether would work.

Make all three colours equally likely

Can you place red, green and blue balls in the box so that each colour has the same probability of being selected?

What must be true about the number of balls of each colour?

Create an impossible event

Remove every ball of one colour from the box.

The probability of selecting that colour is now 0. The event is impossible.

Create a certain event

Place balls of only one colour in the box.

If a ball is selected, that colour is certain to be chosen. Its probability is 1.

What happens when a ball is not replaced?

This visualiser is particularly useful for thinking about dependent probability.

Suppose a box contains 4 orange balls and 6 purple balls. The probability of selecting an orange ball is initially 4/104/104/10.

Now imagine that an orange ball is selected and not replaced.

The box now contains:

• 3 orange balls
• 6 purple balls
• 9 balls altogether

The probability of selecting orange on the second choice has changed from 4/104/104/10 to 3/93/93/9.

The result of the first selection has affected the probability of the second selection. These are dependent events.

If the first ball had been replaced, the box would still contain 4 orange and 6 purple balls. The probabilities would remain unchanged.

This difference between with replacement and without replacement is an important idea when completing probability tree diagrams.

Probability trees and dependent events

A probability tree shows the possible outcomes of two or more events.

When an object is selected without replacement, the probabilities on the second set of branches may be different depending on what happened first.

For example, after selecting a red ball, there may be one fewer red ball in the box. After selecting a blue ball, there may instead be one fewer blue ball.

You therefore need to consider each branch of the tree separately.

The dependent probability worksheet gives you self-marking practice with probability trees where objects are selected without replacement.

A useful classroom probability model

Teachers and tutors can use the visualiser to demonstrate probability before moving on to written calculations.

Build a simple box of coloured balls and ask students to predict which colour is most likely to be selected. Change one ball at a time and discuss how the probabilities change.

The tool can also be used to introduce:

• probabilities written as fractions
• complementary probabilities
• impossible and certain events
• equally likely outcomes
• experimental questions about changing sample spaces
• dependent probability and selection without replacement

The aim is simple: change the model, make a prediction, and explain why the probability has changed.