Given a non uniform 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 15with 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<Npoints each with the probability N^-1. Let the nonuniform sample space have n points, the first point corresponding to N₁ points of the uniform space, the second to N₂ points, etc. What is N₁ + N₂ + .... N_n ?What are p₁, p₂, ...the probabilities associated with the first, second, etc., points of the nonuniform space? What is p₁ + p₂ +....+ p_n? Now consider an event for which several points, say i, 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 15as a specific example if you need to.

Step by step solution

01

Given Information

Non-uniform sample space and the probability associated with the points.

02

Definition of Probability

Probability is a metric for determining the possibility of an event occurring.

03

Prove the statement.

Set S contains N points with equally likely probability,

S = {N₁, N₂, N₃, ..... N_N}

Let A be the event where set A is the part of the mother set S,

A = {N₁, N₂, N₃, ..... N_n}

The Probability of set A is given as follows.

p(A) = p{N₁+ N₂ + N₃+ ..... + N_n}

p(A) = p(N₁) + p(N₂) + p(N₃) + ..... + p(N_n)

p(A) = (1/N)₁ + (1/N)₂ + ..... + (1/N)_n

p(A) = (n/N)

Let R be an event that, in turn, is a subset of S. The probability of points is given below,

R = {N_i, N_j, N_k}

The probability of each point is 1/N , the probability of R is given below,

p(R) = p{N_i, N_j, N_k}

p(R) = p(N_i) +p(N_j) +p(N_k)

p(R) = 1/N + 1/N + 1/N = 3/N

p(R) = 3/N

Hence, the values are mentioned below,

p(A) = (n/N)

p(R) = 3/N

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