Some Instructive Problems
P robability lends itself to a variety of misinterpretations and misunderstandings. Few people would have difficulty understanding, at least in a practical sense, such statements as “The probability that a toss of a fair die will result in a 5 is 1/6, ” and “The probability that the children in a five-child family are all boys is 1/32. ” It is easy to describe fairly simple probabilistic situations, however, that can confuse, at least momentarily, even people with a considerable degree of mathematical sophistication. Usually careful reflection on the situation suffices to clarify it. The first few problems described in what follows are of this type.
There are also probabilistic reasoning problems on which people who are very well tutored in statistics and probability theory have been known to disagree. The last several problems described herein are representative of those that may be in this category. It is my belief that the difficulties usually arise because of a lack of sufficient clarity and precision in the use of language and can be resolved by a careful consideration of what could be meant by the problem statement and how it is interpreted by each of the parties who disagree. Several of these problems are discussed also by Falk (1993).
Imagine the following gamble. A coin is to be tossed repeatedly until one of the two following sequences occurs: (A) head, head or (B) tail, head. Depending