Probability of snow on a given day?

Question

The probability it snows on a given day is 0.3. Hence the probability it won't snow is 0.7. Consider the next 3 days. The probability it does not snow on all 3 days is $0.7^3$. However the probability it snows on exactly one day is $0.3 \times 0.7^2\times3$.

Somehow the probability of snow on a single day of the next three is more likely than it not snowing on all 3 days. This seems extremely unintuitive as the probability of it snowing is less likely than it not snowing. So what's going on here?

Answer 1

suppose you have a dice. the probability of getting 1 or 2 is 1/3 and the probability of not getting them therefore is 2/3. so on each roll it's more likely that you don't get 1 or 2. but what if you throw it 10 times. is it really more likely to not get 1 or 2 at all than to get like 3 or 4 times and to fail 7 or 6 times? wouldn't you feel unlucky if you rolled the dice and didn't get 1 or 2 for like 10 rolls in a row?

Answer 2

What is going on here is that you compare "the sum of three probabilitied of events" with "the probability of three events".

Note that the "$\times 3$" accounts for the three cases (SNN), (NSN) and (NNS) where S=snowing and N=not snowing.

Answer 3

This seems extremely unintuitive as the probability of it snowing is less likely than it not snowing.

Yes, it seems "unintuitive", because it assumes the probability to snow is $0.3$, which is too large. In reality, the chance to snow is very small, maybe around $0.01\sim0.02$. You can imagine how many days snowed in the past year and divide $365$ days as a rough idea how small the chance of snowing should be. So if you set the chance of snowing to be the real-world chance $p=0.02$

The probability of never snow in three days is: $0.98^3=0.942$

The probability of exactly snow on one day is: $3\cdot 0.02\cdot 0.98^2=0.058$

This will agree with our common sense.