Experimental Probability, Part I

Experimental probability is usually introduced to students somewhere between the fifth and seventh grades and could be defined as follows:

In the context of a probability experiment, the experimental probability of an event is the ratio of the observed outcomes in the event to the total number of trials in the experiment.

Suppose a person rolls a number cube (a die) 50 times. In that case, rolling the die is the probability experiment, and the total number of trials in the experiment is 50. The table below shows possible results of that probability experiment:


So, consider the event "rolling an even number." According to the probability experiment results shown above, there were 24 outcomes observed in this event—the person rolled a two 7 times, a four 8 times, and a six 9 times: 7 + 8 + 9 = 24. Thus, one could say that the experimental probability of the event "rolling an even number" is 24/50, or 12/25, or 0.48, or 48%.

Now consider the event "rolling a 2." According to the probability experiment results shown above, there were 7 outcomes observed in this event—the person rolled a two 7 times. Thus, one could say that the experimental probability of the event "rolling a 2" is 7/50, or 0.14, or 14%.

When it is introduced, experimental probability is usually distinguished from theoretical probability, which could be defined as follows:

In the context of a probability experiment, the theoretical probability of an event is the ratio of the number of possible outcomes in the event to the total number of possible outcomes in the experiment.

Consider again the probability experiment described above—rolling a number cube, or die—and consider the first event we discussed above, "rolling an even number." The number of possible outcomes in that event is 3—there are 3 ways to roll an even number: by rolling a 2, 4, or 6. And the total number of possible outcomes in the experiment is 6—the person could roll a 1, 2, 3, 4, 5, or 6. Thus, one could say that the theoretical probability of the event "rolling an even number" is 3/6, or ½, or 0.50, or 50%.

Now consider again the event "rolling a 2." The number of possible outcomes in that event is 1—there is 1 way to roll a 2—and the total number of possible outcomes in the experiment is 6. Thus, one could say that the theoretical probability of the event "rolling a 2" is 1/6, or about 16.7%.

The key differences to note, at least for this post, between the concepts of experimental and theoretical probability as they are introduced to students are that where experimental probability employs "number of observed outcomes in an event," theoretical probability employs "number of possible outcomes in an event," and where experimental probability uses "total number of trials in an experiment," theoretical probability uses "total number of possible outcomes in an experiment." Obviously, these different definitions give us different probabilities for the same events. In the context of the probability experiment described above, the experimental probability of rolling an even number is 48%, whereas the theoretical probability of the same event is 50%. The experimental probability of rolling a 2 is 14%, whereas the theoretical probability of the same event is about 16.7%.

The Argument

Given that background information, consider the following argument put to me a few months ago, in reference to a word problem like the following:

A bag contains ten tiles numbered 1–10. Megan chooses a tile from the bag, records the number on it, and then replaces it seven times. She chooses the number 4 twice, the number 6 once, the number 2 three times, and the number 5 once. Based on these results, what is the experimental probability that the next number Megan chooses will be a 2?

Quite simply, the argument was this: Experimental probability can not be used to discuss the probability of a future event such as "the next number Megan chooses" because experimental probability uses "total number of trials in an experiment," and, for future events, there are no trials to consider.

Well, such an idea contradicts at least one published source:

Megan plays on the high school's varsity softball team. She has been at bat 35 times this season. She gets a hit 9 times. What is the experimental probability that she gets a hit her next time at bat?

So, what do you think?

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