Module: Basic statistics
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For reference, here is a list of the rules of probability:
It is important to bear in mind the circumstances in which the special addition rule and the special multiplication rule can be used. Most mistakes in probabilistic reasoning occur because someone assumes that events are independent when they are not (or vice versa), or because someone assumes that events are mutually exclusive when they are not (or vice versa).
Here is the original problem which led Pascal and Fermat to develop probability theory:
Suppose you roll a single die four times; what is the probability of rolling at least one 6? The gambler reasoned that since the chance of a 6 in each roll is 1/6, the chance of a 6 in 4 rolls is 4 x 1/6 = 2/3. Now suppose you roll a pair of dice 24 times; what is the probability of rolling at least one double 6? The gambler reasoned that since the chance of a double 6 in one roll is 1/36, the chance of a double 6 in 24 rolls is 24 x 1/36 = 2/3. In other words, the gambler expected to win each bet 2/3 of the time. His problem was that he seemed to lose more often with the second bet than the first. He was at a loss to explain this, so he asked his friend Pascal for an answer.
What are the mistakes in the gambler's reasoning? What are the true probabilities of winning each bet?
A fallacy is a mistake in reasoning. The following examples each contain some reasoning about probabilities, some of which is correct and some of which is mistaken. See if you can spot any mistake, and then click "correct" if you think the reasoning is o.k. and "fallacy" if you think it is wrong.