Theoretical Versus Experimental Probability

Student: I just flipped a coin several times and the coin landed on heads more than tails. I had 9 heads and only 6 tails. I don't understand why that happened since heads and tails should be equally likely.

Mentor: How do you know that landing on heads is just as likely as landing on tails when a coin is tossed?

Student: Well, a coin only has two sides (heads and tails) so that means that flipping a coin can have two possible outcomes. The chances for both are equal since the coin is essentially the same on both sides. Therefore, the chances of a coin landing on heads would be 1/2, and the chances of it landing on tails would also be 1/2.

Mentor: That is right! What you just found was the theoretical probability of a coin landing on heads (1/2 or 50%) and a coin landing on tails (1/2 or 50%).

Student: Alright, well why aren't my results from flipping a coin just now the same as the theoretical probability of flipping a coin?

Mentor: Theoretical probability is a way of estimating what could happen based on the information that you have; it is a calculation. Theoretical probability cannot predict what the actual results will be, but it does give you an idea of what is likely to happen in a situation.

Student: I understand that. So the results of flipping a coin should be somewhere around 50% heads and 50% tails since that is the theoretical probability.

Mentor: Yes! Now let's look at the coin flipping game that you just played. What were the results?

Student: The coin landed on heads 9 times and on tails 6 times. That means I flipped the coin 15 times.

Mentor: OK, we are going to use this information to find another form of probability called experimental probability. To find the experimental probability, you find the ratio of the number of trials with a certain outcome to total number of trials. Experimental probability of winning= # of trials with a certain outcome/# of total trials. So let's first find the experimental probability of flipping heads. For this situation the number of games won would be the number of flips that landed on heads. That would be 9.

Student: The number of games played was 15, so that means that the experimental probability is 9/15 (or simplified, 3/5)!

Mentor: Now what would the experimental probability of flipping tails be?

Student: Well, the number of games won in this situation would be the number of times that I flipped tails, so 6. Then, I played 15 games so the ratio would be 6/15 (or simplified, 2/5).

Mentor: Great. Now, can you write these experimental probabilities as percents?

Student: I would multiply 3/5 by 100% and get 60% as the experimental probability of flipping heads. Then for tails I would multiply 2/5 by 100% and get 40%. The experimental probability of flipping tails is 40%.

Mentor: The experimental probabilities were 40% tails and 60% heads. This does not precisely match with the theoretical probability of 50% tails and 50% heads. However, they are not too far off. Let's do an experiment! Using the coin toss activity, toss the coin 25 times and then 150 times.

Student: OK, after 25 tosses I got 11 heads and 14 tails, and after 150 tosses I got 71 heads and 79 tails.

Mentor: Alright, we know the theoretical probability will be 50% heads and 50% tails no matter how many trials, but what would the experimental probability be in this experiment?

Student: For 25 tosses the probability of heads would be 11/25 (44%) and for tails would be 14/25 (56%). For 125 tosses the probability of heads would be 71/150 (about 47%) and the probability of tails would be 79/150 (about 53%).

Mentor: Now which results have the experimental probability closer to the theoretical probability?

Student: After 25 tosses, the experimental probabilities of heads and tails are not very close to 50%. However, after 150 tosses the experimental probabilities for heads and tails are much closer to 50%.

Mentor: Can you make an educated guess at what that means?

Student: Well, it seems that with more tosses, the resulting experimental probabilities are closer to the theoretical probabilities.

Mentor: Good job! As the amount of trials (in this case a trial is flipping a coin) increases, the experimental probability gets closer to the theoretical probability. You can test this concept with the Crazy Choices Game.