Module 6: Probability and Probability Distributions
Module 6: Probability and Probability Distributions
Another Look at Probability (2 of 2)
Another Look at Probability (2 of 2)
Learning OUTCOMES
- Interpret (in context) a probability as a long-run relative frequency of an event.
What Is the Relationship between Theoretical and Empirical Probability?
We investigate this question in the following two activities. We use coin flipping as a first step in understanding the connection between these two ways of determining the probability of an event.
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 10 times, we are not guaranteed to get 5 heads and 5 tails. So what exactly does it mean when we say P(heads) = 0.5? To answer this question, we use a simulation to simulate flipping a coin.
Our goal is to understand how the empirical probability P(head) relates to the theoretical probability of 0.5.
Activity 1: Fair Coin
The purpose of this activity is to experiment with a simulation that simulates flipping a fair coin, and to see if the P(H) = 0.5.
Source: GeoGebra, license: CC BY SA
Part (1)
- Make sure Coins = 1 and P(heads) = 0.5.
- 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.
Part (2)
- Press the Reset button so that the count is cleared.
- Make sure Coins = 1 and P(heads) = 0.5.
- This time press the “10 Flips” button 3 times so that you have 30 coin flips.
Try It
Part (3)
- Press the Reset button so that the count is cleared.
- Make sure Coins = 1 and P(heads) = 0.5.
- Press the Auto button and watch the count of heads and tails change.
- Click the Pause (II) button once Total Flips is over 1,000.
Try It
Try It
In the preceding activity, the simulation simulates flipping a fair coin. P(heads) = 0.5 with a fair coin. How can we tell if a coin is not fair? Theoretical probability methods cannot answer this question. The only way we can answer this question is to collect data as we flip the coin.
Activity 2: Unfair Coin
The purpose of this activity is to experiment with an activity that simulates flipping an unfair coin.
Source: GeoGebra, license: CC BY SA
- Make sure Coins = 1 and P(heads) = 0.2.
- Click the Auto button and watch the count of heads and tails change.
- Click the Pause (II) button once Total Flips is over 100 or so.
- Record the total number of Heads (1’s) and the total number of flips.
- Calculate P(H) (Number of heads / Total Flips) when Total Flips is about 100.
- Click the Auto button again to continue the flips.
- Click the Pause (II) once Total Flips is over 1,000 or so.
- Record the total number of Heads (1’s) and the total number of flips.
- Calculate P(H) (Number of heads / Total Flips) when Total Flips is about 1,000.
Try It
Try It
Let’s summarize what we have learned from these activities:
- The empirical probability will approach the theoretical probability after a large number of repetitions. In some situations, such a