Chapter 15.2, Problem 16P

Given an nonuniform sample space and the probabilities associated with the points, we defined the probability of an event $A$ as the sum of the probabilities associated with the sample points favorable to $A$.

[You used this definition in Problem 15 with the sample space ( 2.5 ).] Show that this definition is consistent with the definition by equally likely cases if there is also a uniform sample space for the problem (as there was in Problem 15 ). Hint: Let the uniform sample space have $N$ points each with the probability $N^{-1}.$ Let the nonuniform sample space have $n<N$ points, the first point corresponding to $N_1$ points of the uniform space, the second to $N_2$ points, etc. What is $N_1+N_2+\cdots +N_n?$

What are $p_1,p_2,\dots ,$ the probabilities associated with the first, second, etc., points of the nonuniform space? What is $p_1+p_2+\cdots +p_n?$ Now consider an event for which several points, sayi, j, k, of the nonuniform sample space are favorable. Then using the nonuniform sample space, we have, by definition of the probability $p$ of the event, $p=p_i+p_j+p_k.$ Write this in terms of the $N$ ’s and show that the result is the same as that obtained by equally likely cases using the uniform space. Refer to Problem 15 as a specific example if you need to.