Question

Let \(S=\{1,2,3,\ldots\}\) and suppose that every subset of \(S\) is an event. Let \(\mathcal{P}(S)\) denote the power set of \(S\). Which of the following statements is/are true?

• A. There exists a probability function \(P:\mathcal{P}(S)\to [0,\infty)\) such that \(P(\{n\})=\dfrac{1}{n+1},\; n\geq 1\)
• B. There exists a probability function \(P:\mathcal{P}(S)\to [0,\infty)\) such that \(P(A)=0\) if \(A\) is a finite set and \(P(A)=1\) if \(A\) is an infinite set
• C. There exists a probability function \(P:\mathcal{P}(S)\to [0,\infty)\) such that \(P(\{1,2,\ldots,n\})=\int_{1}^{n}\dfrac{1}{x}\,dx,\; n\geq 1\)
• D. There exists a probability function \(P:\mathcal{P}(S)\to [0,\infty)\) such that \(P(\{1,2,\ldots,n\})=\int_{0}^{n}e^{-x}\,dx,\; n\geq 1\)

Hint:

For a probability measure on a countable set, the probabilities assigned to all singleton elements must be non-negative and their total sum must be \(1\).

Answer 1

The Correct Option is D

Step 1: Test (A).
If $P(\{n\})=\frac1{n+1}$, then $\sum_n\frac1{n+1}$ diverges, so the total cannot be $1$. (A) fails.

Step 2: Test (B).
Evens and odds are both infinite, so each would get probability $1$, summing to $2$ for the whole space. Impossible, so (B) fails.

Step 3: Test (C).
$P(\{1,\dots,n\})=\int_1^n\frac1x dx=\ln n$, which exceeds $1$ for large $n$. A probability cannot do that, so (C) fails.

Step 4: Test (D).
$P(\{1,\dots,n\})=\int_0^n e^{-x}dx=1-e^{-n}$ stays in $[0,1)$ and rises to $1$. The singleton masses $e^{-(n-1)}-e^{-n}$ are non-negative and sum to $1$, so this works. (D) holds.

Step 5: Collect.
\[ \boxed{(D)} \]