Wiki Page, Math 13 - Perez Sandbox : Mod 11 Probability Applets Lab Activity

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2026-01-17T21:15:05-08:00

What Is the Relationship between Theoretical and Empirical Probability?

We investigate this question in the following two activities using coin flipping.

You will not need to turn anything in for this lab. Just work through the directions and questions below to understand the connection between theoretical probability and empirical probability.

Purpose

Our goal is to understand how the empirical probability relates to the theoretical probability. We will focus on P(heads), the probability of a head. First we will examine what happens with a fair coin. Then we will examine what happens with an unfair coin.

You will use the applet below for both investigations. Follow the instructions posted below the applet.

GeoGebra applet (CC BY SA)

Part (1): A Fair Coin

A single flip of a coin has an uncertain outcome. We do not know if we will get heads or tails. If we flip the coin 6 times, we are not guaranteed to get 3 heads and 3 tails. So what exactly does it mean when we say the theoretical probability P(heads) = 0.5? To answer this question, we use an applet to simulate flipping a fair coin.

1. Start by setting the applet to represent the flipping of a fair coin. Set P(heads) to 0.5.

2. Now use the applet to simulate flipping a fair coin three times.
Set Coins to 1. Press the “1 Flip” button 3 times.
Notice that for each flip, you will see either heads (1) or tails (0) appear in the histogram count. Obviously, these are the only possibilities, so you will not see anything appear for values on the horizontal axis greater than 1 because you are only flipping one coin.

3. First estimate of the theoretical probability P(heads) for a fair coin using 3 flips.
Based on your 3 flips, what is the estimate for P(heads)?
(Calculate this by dividing: numbers of heads divided by 3. Why divide by 3? Because there were 3 coin flips.)

4. A second estimate of the theoretical probability P(heads) for a fair coin using 30 flips.
Press the Reset button so that the count is cleared.
Keep Coins = 1 and P(heads) = 0.5. Now press the “10 Flips” button 3 times so that you have 30 coin flips. (This is easier than pressing "1 Flip" thirty times!)

1. Based on your 30 flips, what is your estimate for P(heads)?
(Calculate P(heads) = numbers of heads / 30)
How close was your estimate to the theoretical probability of 0.5?

2. Obviously, we will not always get 15 heads out of 30 flips. The histogram and table below show results for 12 of your classmates. Each small rectangle represents an empirical estimate for P(heads) from 30 flips.

To check your understanding of this graph, identify where your P(heads) estimate would be in the histogram.

Results of flipping 30 coins
with P(heads) = 0.5

Person | # of heads | proportion
1 | 15 | 0.50
2 | 17 | 0.57
3 | 14 | 0.47
4 | 15 | 0.50
5 | 14 | 0.47
6 | 19 | 0.63
7 | 12 | 0.40
8 | 14 | 0.47
9 | 20 | 0.67
10 | 9 | 0.30
11 | 10 | 0.33
12 | 16 | 0.53

5. A third estimate of the theoretical probability P(heads) for a fair coin using 1000 flips.
Press the Reset button so that the count is cleared.
Keep Coins = 1 and P(heads) = 0.5. Now press the Auto button and watch the count of heads and tails change. Click the Pause button once Total Flips is 1,000.

1. Based on your 1000 flips, what is your estimate for P(heads)?

2. Obviously, we will not always get 500 heads out of 1000 flips. The histogram and table below show results for 12 of your classmates. Each small rectangle represents an empirical estimate for P(heads) from 1000 flips.

Results of flipping 1000 coins
with P(heads) = 0.5

Person | # of heads | proportion
1 | 493 | 0.49
2 | 497 | 0.50
3 | 500 | 0.50
4 | 500 | 0.50
5 | 510 | 0.51
6 | 504 | 0.50
7 | 520 | 0.52
8 | 499 | 0.50
9 | 505 | 0.51
10 | 523 | 0.52
11 | 514 | 0.51
12 | 483 | 0.48

Out of 12 people, only two people got an estimate for P(heads) that was exactly 0.5 (500 heads in 1000 flips.) Results ranged from 483 to 523 heads, so estimates for P(heads) ranged from about 0.48 (483 out of 1000) to 0.52 (523 out of 1000). The average empirical probability estimate (shown by the green triangle in the histogram) was 0.504, which is very close to the theoretical probability.

6. Which provides a better estimate of the theoretical probability P(heads) for the fair coin: an empirical probability using 30 flips or 1000 flips? Why do you think so?

Part (2): A Unfair Coin

1. Start by setting the applet to represent the flipping of an unfair coin. Set P(heads) to 0.2 to represent a coin that has only a 20% probability of landing on heads. We will repeat some of Part 1 using the unfair coin to examine how empirical probability estimates relate to the theoretical probability of 0.2.

2. An estimate of P(heads) for an unfair coin using 30 flips.
With Coins = 1 and P(heads) = 0.2, press the “10 Flips” button 3 times so that you have 30 coin flips. (This is easier than pressing "1 Flip" thirty times!)

1. Based on your 30 flips, what is your empirical estimate for P(heads)?
How close was your estimate to the theoretical probability 0.2?

2. The table and histogram below show results for 12 of your classmates. Each small rectangle represents an estimate for P(heads) from 30 flips.
To check your understanding of this graph, identify where your P(heads) estimate would be in the histogram.

Results of flipping 30 coins
with P(heads) = 0.2

Person | # of heads | proportion
1 | 8 | 0.27
2 | 9 | 0.30
3 | 3 | 0.10
4 | 8 | 0.27
5 | 8 | 0.27
6 | 8 | 0.27
7 | 9 | 0.30
8 | 3 | 0.10
9 | 6 | 0.20
10 | 10 | 0.33
11 | 5 | 0.17
12 | 6 | 0.20

Two of the 12 students got exactly 6 heads, which is 6/30=0.2. But many students did not. Results ranged from 3 to 10 heads, so estimates for P(heads) ranged from 0.10 (3 out of 30) to 0.33 (10 out of 30) with an average of 0.231 (6.93 heads out of 30), which is close to the theoretical probability of 0.2.

3. Another estimate of P(heads) for an unfair coin using 1000 flips.
Press the Reset button so that the count is cleared.
Keep Coins = 1 and P(heads) = 0.2. Now press the Auto button and watch the count of heads and tails change. Click the Pause button once Total Flips is 1,000.

1. Based on your 1000 flips, what is your empirical estimate for P(heads)?

2. Obviously, when the theoretical probability is 0.20, we will not always get 200 heads out of 1000 flips. The histogram below shows results for 12 of your classmates. Each small rectangle represents an estimate for P(heads) from 1000 flips.

Results of flipping 1000 coins
with P(heads) = 0.2

Person | # of heads | proportion
1 | 190 | 0.19
2 | 177 | 0.18
3 | 221 | 0.22
4 | 185 | 0.19
5 | 203 | 0.20
6 | 214 | 0.21
7 | 206 | 0.21
8 | 192 | 0.19
9 | 206 | 0.21
10 | 192 | 0.19
11 | 198 | 0.20
12 | 201 | 0.20

Out of 12 people, no one got exactly 200 heads in 1000 flips. Results ranged from 177 to 221 heads, so estimates for P(heads) ranged from about 0.18 (177 out of 1000) to 0.22 (221 out of 1000) with an average of 0.199 (199 heads), which is very close to the theoretical probability of 0.200.

4. Which provides a better estimate of theoretical probability P(heads) for the unfair coin: an empirical probability using 30 flips or 1000 flips? Why do you think so?

What Can We Learn From These Activities?

Here are the big conceptual points:

This activity helped us examine the relationship between empirical probability and theoretical probability.

An empirical probability is an estimate based on a relative frequency: a calculation of how often an outcome occurs in a long series of repetitions. It is based on data.

A theoretical probability is what we expect to happen. It is not based on data.

When we examine empirical probabilities, on average they do a good job estimating a theoretical probability.

There is less variability in the empirical probabilities with a large number of repetitions.

In some situations we will not know, or it is impossible to determine, a theoretical probability, so we can use a single empirical probability as an estimate. In these situations, we are more confident about estimating the probability of an outcome using an empirical probability with a large number of repetitions.