I'm trying to calculate the probability of a natural number being divisible by 2, 3, or 5 and I feel as if I may have found the answer. But I wanted to see if anyone sees anything wrong with my "work". Thank you all for your time and help.
Let ~ signify 'n is divisible by':
P[~2 ∨ ~3] = P[~2] + P[~3] – P[~2 ∧ ~3] = 1/2 + 1/3 – 1/6 = 2/3
P[(~2 ∨ ~3) ∨ ~5] = P[~2 ∨ ~3] + P[~5] – P[(~2 ∨ ~3) ∧ ~5] = 2/3 + 1/5 – something
something = P[(~2 ∨ ~3) ∧ ~5] = P[(~2 ∧ ~5) ∨ (~3 ∧ ~5)] = 1/10 + 1/15 – 1/30 = 4/30 = 2/15 so
P[(~2 ∨ ~3) ∨ ~5] = 2/3 + 1/5 – 2/15 = 11/15
Are these calculations correct and am I even using probabilities and such correctly?
Best Answer
You have to describe how you randomize a natural number. It is not possible to have a discrete uniform distribution on $\mathbb{N}$. If you are talking about the natural density of natural numbers divisible by $2$, $3$, or $5$, then the answer is $\frac{11}{15}$. Alternatively, if you are talking about the discrete uniform distribution on $\mathbb{Z}/30\mathbb{Z}$, then your calculation is correct.