Bag of marbles and rocks

If I first pulled a rock out of the bag, what would the probability be of pulling out a marble next?

If I first pulled a marble out of the bag, what would the probability be of pulling out another marble next?

Press ‘Answer’ to check the explanation.

Answer

Task 1: Dependent and independent events

The marble and rock experiment is an example of a dependent event. Since we are removing items from the bag, the next event depends on what happened in the previous event.

When trying to determine the probability of events, we typically work with independent events. These are events where the probability is not affected by previous events. For example, when given two options A and B, the probability of selecting one option will always be $\frac{1}{2}$ or 50%. If I get option A in my first selection, it will not change the probability of whether or not I will also get option A in my next selection. The probability for option B will remain $\frac{1}{2}$ or 50%. What happened previously will not affect the current event.

Think back to the marble and rock experiment. How could we change the experiment so that it is an independent event instead?

Throughout this learning activity, you can record your ideas digitally, orally, or in print.

Press ‘Answer’ to check the answer.

Answer

If we replace the items in the bag each time, then the probabilities do not change and the events are independent:

• With replacement, the events are independent (the probabilities don’t change).

• Without replacement: the events are dependent (the probabilities change).

Task 2: Independent or dependent?

For the following events, indicate whether they are independent or dependent.

Press 'Check Answer' to check how you did.

Tree diagram of dependent events

Tree diagrams are useful for keeping track of independent events. They are also helpful in keeping track of the probability of dependent events.

Let’s revisit the marbles and rocks in the bag from the Minds On section.

Examine the branches of the tree diagram and label the diagram as explained.

There are 5 items in the bag: 2 marbles and 3 rocks.

There is a $\frac{2}{5}$ chance of pulling out a marble, and a $\frac{3}{5}$ chance for a rock.

We can go one step further and examine what happens when we pick a second item:

• If a marble was selected first, there is now a $\frac{1}{4}$ chance of getting a marble and a $\frac{3}{4}$ chance of getting a rock.
• If a rock was selected first, there is now a $\frac{2}{4}$ chance of getting a marble and a $\frac{2}{4}$ chance of getting a rock.

Organize your data by labelling the branches, or using columns, categories, or a list.

Press ‘Tree Diagram’ to check your answer.

Tree Diagram

Now, we can answer questions like, “What are the chances of selecting 2 marbles?.” In a tree diagram, to calculate the probability, we multiply along the branches. The top branch, has one marble being selected (which is $\frac{2}{5}$ items) then one more marble (now only $\frac{1}{4}$ items).

It is a $\frac{2}{5}$ chance followed by a $\frac{1}{4}$ chance

We can multiply the probabilities of the two dependent events to see what the overall probability is of drawing 2 marbles. Calculate the probability and check your answer by pressing the button ‘Probability.’

Probability

It's up to you

Select an activity of your choice to collect probability data about. You might choose to roll dice or select cards. It is entirely up to you.

Create a tree diagram or use columns or categories to calculate the probability of drawing different outcomes.

Record your ideas digitally, on paper, or as a concrete model.

Reflection

As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.

Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.

When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.

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