# Tossing heads - Classical and Empirical Probabilities

by Aleshia

(Mount Gay, West Virginia)

**Classical and Empirical probabilities** • Contrast and compare classical and empirical probabilities

a) Explain the two major differences between the two

• Gather data on empirical probabilities by completing the following experiment

1. Collect an EVEN NUMBER of coins (any denomination).

a) the total number of coins should be between 16 and 30 coins

2. Place all of your coins in a container which is large enough to mix the coins together when it is shaken.

3. Shake the bag well and empty the coins onto a table.

4. Record how many coins have the head side up, and how many coins have the tail side up.

a) Organize the results of your experiment in a table or chart.

b) Be sure to list how many coins you have.

5. Repeat steps 3 and 4 nine more times. This will allow you to record the results of ten experiments.

• Analyze the results of your experiment

1. Look at the data from your first count of the tossed coins.

a) What was the observed probability of tossing a head?

b) What was the observed probability of tossing a tail?

c) Show the formula you used and reduce the fractions (which represent the probability) to their lowest terms.

2. Did any of your ten repetitions come out to have exactly the same number of heads and tails?

a) If so, how many times did this happen?

3. Why are the probabilities for step 2 not (exactly) ½ and ½ for each of the ten trials?

4. What kind of probability is represented by this “bag of coins” experiment?

5. Compute the average number of heads from the ten trials (add up the number of heads and divide it by 10).

6. Change this to the average probability of tossing heads by putting the average number of heads in a fraction over the number of coins you used in your tosses.

7. Did anything surprising or unexpected happen in your results for this experiment?

8. Write the sample space for the outcomes of tossing three coins using H for heads and T for tails.

9. What is the calculated probability for each of the outcomes?

10. Which kind of probability did you calculate in step 9 (theoretical or empirical)?

11. Why don’t you need to use physical coins to compute the probabilities for these outcomes?