Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/269

 

CHAPTER XVII.

DEMONSTRATION OF A GENERAL METHOD FOR THE SOLUTION OF PROBLEMS IN THE THEORY OF PROBABILITIES.

1. IT has been defined (XVI. 2), that "the measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or unfavourable, and all equally possible." In the following investigations the term probability will be used in the above sense of "measure of probability."

From the above definition we may deduce the following conclusions.

1. When it is certain that an event will occur, the probability of that event, in the above mathematical sense, is $\scriptstyle {1}$. For the cases which are favourable to the event, and the cases which are possible, are in this instance the same.

   Hence, also, if $\scriptstyle {p}$ be the probability that an event $\scriptstyle {x}$ will happen, $\scriptstyle {1-p}$ will be the probability that the said event will not happen. To deduce this result directly from the definition, let $\scriptstyle {m}$ be the number of cases favourable to the event $\scriptstyle {x}$, $\scriptstyle {n}$ the number of cases possible, then $\scriptstyle {n-m}$ is the number of cases unfavourable to the event $\scriptstyle {x}$. Hence, by definition,

   $\scriptstyle {{\frac {m}{n}}=}$probability that $\scriptstyle {x}$ will happen.

   $\scriptstyle {{\frac {n-m}{n}}=}$probability that $\scriptstyle {x}$ will not happen.

   But$\scriptstyle {{\frac {n-m}{m}}=1-{\frac {m}{n}}=1-p}$.

2. The probability of the concurrence of any two events is the product of the probability of either of those events by the probability that if that event occur, the other will occur also.

   Let $\scriptstyle {m}$ be the number of cases favourable to the happening of the first event, and $\scriptstyle {n}$ the number of equally possible cases unfavourable to it; then the probability of the first event is, by defini-