# Lesson 4: Estimating Probabilities Through Repeated Experiments

Let’s do some experimenting.

## 4.1: Decimals on the Number Line

1. Locate and label these numbers on the number line.

1. 0.5
2. 0.75
3. 0.33
4. 0.67
5. 0.25

2. Choose one of the numbers from the previous question. Describe a game in which that number represents your probability of winning.

## 4.2: In the Long Run

Mai is playing a game where she will win only if she rolls a 1 or a 2 with a standard number cube.

1. List the outcomes in the sample space for rolling the number cube.

2. What is the probability Mai will win the game? Explain or show your reasoning.

3. If Mai is given the option to flip a coin and win if it comes up heads, is that a better option for her to win?

This applet displays a random number from 1 to 6, like a number cube. Mai won with the numbers 1 and 2, but you can choose any two numbers from 1 to 6. Record them in the boxes in the center of the applet.

1. Click the Roll button for 10 rolls, and then answer the questions below.

GeoGebra Applet ZpexxzFy

1. If the roll stops on one of your winning numbers, what happens in the table?

2. What appears to be happening with the points on the graph?

1. After 10 rolls, what fraction of the total rolls were a win?
2. How close is this fraction to the probability that Mai will win?
3. Roll the number cube 10 more times to fill in the table and graph the results, for a total of 20 points on the graph.

1. After 20 rolls, what fraction of the total rolls were a win?
2. How close is this fraction to the probability that Mai will win?

## 4.3: Due For a Win

1. For each situation, do you think the result is surprising or not? Is it possible? Be prepared to explain your reasoning.

1. You flip the coin once, and it lands heads up.
2. You flip the coin twice, and it lands heads up both times.
3. You flip the coin 100 times, and it lands heads up all 100 times.
2. If you flip the coin 100 times, how many times would you expect the coin to land heads up? Explain your reasoning.
3. If you flip the coin 100 times, what are some other results that would not be surprising?
4. You’ve flipped the coin 3 times, and it has come up heads once. The cumulative fraction of heads is currently $\frac{1}{3}$. If you flip the coin one more time, will it land heads up to make the cumulative fraction $\frac{2}{4}$?

## Summary

A probability for an event represents the proportion of the time we expect that event to occur in the long run. For example, the probability of a coin landing heads up after a flip is $\frac12$, which means that if we flip a coin many times, we expect that it will land heads up about half of the time.

Even though the probability tells us what we should expect if we flip a coin many times, that doesn't mean we are more likely to get heads if we just got three tails in a row. The chances of getting heads are the same every time we flip the coin, no matter what the outcome was for past flips.