The following programs were inspired by the paper "Some teasers concerning conditional probabilities" by Maya Bar-Hillel and Ruma Falk (Cognition, 11 (1982) 109-122. The teasers in question "ask for the probability that the objects of a certain pair both have some property when information exists that at least one of them does". These problems can be solved by the application of conditional probability rules, but difficulties arise "in characterizing the conditioning event appropriately". Thus, seemingly simple problems will prompt different responses from different respondants, even if they are well versed in probability theory.
Two programs are presented: Father and Son and Deck of Cards. The essential details of the problems will be discussed. However, for an in-depth examination of the topic, the reader is encouraged to read the paper referenced above.
I've tried to make the UI as intuitive as possible. Parameters affecting the information available in each scenario are changed by using drop-down menus embedded in the problem description.
Sampling is done in one of three ways: Entire Population, Random Sample, or Both. Entire Population means that a large population of samples is generated, and the statistics are generated by considering all of the population members. Random Sample means that a large population of samples is first generated, and then random members of this population are used to generate the statistics. The meaning of Both is self-evident.
The Preferences button displays a pop-up menu that allows you modify the functioning of the program. You may modify both the population and sample sizes here (you can increase the accuracy of the simulation by increasing these values, or speed up program execution by decreasing them). You can also set the random seed used to generate the population. To do this, check the checkbox to the right of User Random Seed, and then enter the value in the input field. Set Preferences makes the changes you requested (if any), and Cancel closes the window without making any changes.
The Run Simulation button does just that. Output is displayed on a pop-up window that can be dismissed by clicking its OK button. Many of these output windows can be open at once. The output values are the percentages of the stipulated condition occuring in the sample and/or population.
Netscape's Java VM sometimes has difficulty with the programs. Make sure that once you click one of the "Start applet" buttons, you run at least one simulation before closing its window. (I have no idea why this works.)
Here is the first version of the problem as presented by Bar-Hillel and Falk:
Mr. Smith is the father of two. We meet him walking along the street with a young boy whom he proudly introduces as his son. What is the probability that Mr. Smith's other child is also a boy?
and here is the second:
We meet Mr. Smith (whom we know to be the father of two) in the street with a boy. This time, he is more elaborate in his introduction, presenting the boy as his eldest child. What is the probability that Mr. Smith's other child is also a boy?
Notice how the resulting probabilities change in the second version of the problem. For an explanation of the differences, see the paper referenced in the Introduction.
This is a similar puzzle, formulated in terms of a card problem. Again, there are two versions. Here is the first:
A deck of four cards consists of the ace of spades, the ace of clubs, the deuce of spades, and the deuce of clubs. A hand of two cards is randomly dealt from this deck. What is the probability that it contains both aces if we know it contains at least one?
The second version is similar; only the last sentence is changed:
What is the probability that the had contains both aces if we know it contains the ace of spades?
Again, notice the differences when the simulation is run.
These programs were written by Joshua Gargus (as a summer student) for Jeff Pelletier in August, 1997