[T1.6] Statistical reasoning

Summary: the rules of probability

For reference, here is a partial list of the rules of probability:

Addition rule:

$\begin{displaymath}P(A \mbox{ or } B) = P(A) + P(B) - P(A \mbox{ and } B) \end{displaymath}$
Special addition rule (for mutually exclusive events):

$\begin{displaymath}P(A \mbox{ or } B) = P(A) + P(B) \end{displaymath}$
Subtraction rule:

$\begin{displaymath}P(\mbox{not } A) = 1 - P(A) \end{displaymath}$
Multiplication rule:

$\begin{displaymath}P(A \mbox{ and } B) = P(A\vert B)P(B) \end{displaymath}$
Special multiplication rule (for independent events):

$\begin{displaymath}P(A \mbox{ and } B) = P(A)P(B) \end{displaymath}$

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).

Self-test question:

Read the original problem which led Pascal and Fermat to develop probability theory. What are the mistakes in the gambler's reasoning? What are the true probabilities of winning each bet?