Melissa Jackson and Kady Schneiter
Utah State University

The Probability Aquarium is a Java applet that presents basic probability rules in the context of interactive questions based on selecting fish at random from an aquarium. For each of eight different questions increasing in complexity, the student performs sampling with or without replacement and then answers a probability question based on this experiment.

The questions for student investigation are included beneath the applet and also on the second page of this posting for easy viewing and printing.

Please note that the Java archive file is about 3MB, so this applet takes a minute or two to load and initialize.

Important note: Some versions of the Java browser plug-in have some difficulty loading the above applet from the MathDL server. If you have any difficulty viewing the applet, there are two ways to fix the problem: you can either download a more recent version of the Java plug-in (we recommend at least version 1.6.0_15) or view the alternate version of the applet (which links to an external copy of the applet, not subject to the loading problem).

Instructions:

• To begin at a different level, enter a number from one to eight in the box and click on 'Go.'
• Select a fish by clicking on or near it.
• Press the 'Done' button to indicate that you have made your selections.
• To answer a probability question, express the probability as a fraction and enter the numerator and denominator in the approrpriate boxes. Then click on 'Submit Answer'.
• If you are unsure of the probability, use the 'Hint' or 'See Solution' buttons to get help.
• From the solution screen, use the buttons to return to the previous level, repeat the current level, or move on to the next level.

The Solution Screen: For each level, the solution screen provides a brief explanation of the probability of the indicated event and a table illustrating how this probability could be found by listing all outcomes.

The Solution Table:

• Each cell in the table corresponds to one way in which the fish could have been selected.
• All possible outcomes are represented in the table.
• The order in which fish are arranged in a cell indicates the order of selection.

If the fish types were chosen randomly, all outcome would have the same probability. Thus the probability that an event occurs can be found by counting the number of outcomes that correspond to that event (these are highlighted in blue) and dividing by the total number of outcomes.

Questions to consider:

• Which fish did you select on Level 1? Be specific.
• Why is the probability of randomly selecting that fish 1/5?
• What is the probability of randomly selecting the type of fish you chose on Level 2?
• What would this probability be if you had chosen a different type of fish?
• On Level 3, how would the probability change if the types of fish were drawn with replacement instead of without replacement?
• On Level 3 and Level 4, did you use the addition rule or the multiplication rule to compute the probability? Explain in your own words why this rule must be used.
• On Level 5, after using the multiplication rule to calculate the probability of randomly selecting the two types of fish you chose in the order you chose them, why must you then double the probability?
• On Level 6, did you use the addition rule to compute the probability of randomly selecting the two fish you chose? Why or why not?
• How do Level 7 and Level 8 differ? How does this affect the probabilities you computed?