EXPERIMENT CONCERNING BERNOULLI-LIKE PROCESS

The Bernoulli process must possess the following properties:
1. The experiment consists of n repeated trials. [Each trial is called a Bernoulli trial.]
2. Each trial results in an outcome that may be classified as a success or a failure.
3. The probability of success, denoted by p, remains constant from trial to trial.
4. The repeated trials are independent.

Now we are going to do an experiment approximating to a Bernoulli process, i.e. Bernoulli-like process. (If you have a deep understanding about the Bernoulli process, you know why the experiment is not an exact Bernoulli process, but it is only an approximation to the process.) We are going to do 20 Bernoulli-like process. In each process, there are 4 Bernoulli trials (n = 4). Ideally, the same single dice is rolled 4 times by the same person. But for the sake of time efficiency, instead of rolling a dice 4 times in each process, each of the 4 students in a group roll a dice simultaneously. Let’s define that a success occurs if a dice shows the side with more than two spots. If a dice shows the side with 1 or 2 spots, we say that a failure occurs. For example, if there are 3 dice each of which shows more than two spots, we say that there are 3 successes and denote it as x = 3. If there is only 1 dice showing the side with one or two spots, we say that there is 1 success and denote it as x = 1. If all the four dice show the sides with more than two spots, we say that there are 4 successes and denote is as x = 4.

Based on the results, fill in the following table.

In Dice #1, Dice #2, Dice #3, and Dice #4 columns, fill the blanks with S if a success occurs and F if a failure occurs.

Then, complete the following table.

The relative frequencies in this case can be considered as empirical probabilities; the corresponding theoretical probabilities can be calculated by applying the concept of binomial distribution (by assuming that the experiment meets the Bernoulli process criteria). Find the theoretical probabilities. Compare these theoretical results with the empirical ones. Explain.

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