practice_problems/problems/unit-01/C-probability-function-properties/2

[giscus[bot]]

practice_problems/problems/unit-01/C-probability-function-properties/2

https://mids-w203.github.io/practice_problems/problems/unit-01/C-probability-function-properties/2.html

[a-sharma06]

I think there is a limits based approach which I found easier to understand (although limits is likely out of scope of this course):

Assume P({k}) = c for all positive integers k.
For any finite N, consider the first N integers: P({1, 2, 3, ..., N}) = P({1}) + P({2}) + ... + P({N}) = Nc

We know, P(Ω) = 1 = lim(N→∞) P({1, 2, ..., N})
But also, lim(N→∞) P({1, 2, ..., N}) = lim(N→∞) Nc

If c>0, lim(N→∞) Nc = ∞ (i.e. not 1)
if c >0, lim(N→∞) N·0 = 0 (i.e. not 1)
if c <0, negative values are not allowed.

There, there is no value P({k}) = c that is the same for every integer.