# Lesson 4

Estimating Probabilities Through Repeated Experiments

### Lesson Narrative

In this lesson students roll a number cube many times and calculate the cumulative fraction of the time that an event occurs to see that in the long run this relative frequency approaches the probability of the chance event. By repeating the experiment and examining the structure of the results, students are engaging in MP8. They also see that the relative frequency of a chance event will not usually exactly match the actual probability. For example, when flipping a coin 100 times, the coin may land showing a head 46 times instead of exactly 50 times and not be considered unreasonable.

In future lessons students will be asked to design and use simulations. Each lesson leading up to that helps prepare students by giving them hands-on experience with different types of chance experiments they could choose to use in their simulations. In this lesson students work with rolling a number cube and tossing a coin.

### Learning Goals

Teacher Facing

• Describe (orally and in writing) patterns observed on a table or graph that shows the relative frequency for a repeated experiment.
• Generalize (orally) that the cumulative relative frequency approaches the probability of the event as an experiment is repeated many times.
• Generate possible results that would or would not be surprising for a repeated experiment, and justify (orally) the reasoning.

### Student Facing

Let’s do some experimenting.

### Required Preparation

The In the Long Run activity requires 1 number cube for every 3 students. Access to graph paper may be useful, but is not required.

### Student Facing

• I can estimate the probability of an event based on the results from repeating an experiment.
• I can explain whether certain results from repeated experiments would be surprising or not.

Building On

### Glossary Entries

• chance experiment

A chance experiment is something you can do over and over again, and you don’t know what will happen each time.

For example, each time you spin the spinner, it could land on red, yellow, blue, or green.

• event

An event is a set of one or more outcomes in a chance experiment. For example, if we roll a number cube, there are six possible outcomes.

Examples of events are “rolling a number less than 3,” “rolling an even number,” or “rolling a 5.”

• outcome

An outcome of a chance experiment is one of the things that can happen when you do the experiment. For example, the possible outcomes of tossing a coin are heads and tails.

• probability

The probability of an event is a number that tells how likely it is to happen. A probability of 1 means the event will always happen. A probability of 0 means the event will never happen.

For example, the probability of selecting a moon block at random from this bag is $$\frac45$$.

• random

Outcomes of a chance experiment are random if they are all equally likely to happen.

• sample space

The sample space is the list of every possible outcome for a chance experiment.

For example, the sample space for tossing two coins is: