Chance of Rain?

Was looking at the weather report this morning, and there’s a probability question that has me stumped…

If there’s a 30% chance of rain on Sunday and a 30% chance of rain on Monday, what are the chances that it rains on EITHER of those days? And by EITHER, I mean one or the other or both.

And what about if there is a 1% chance of rain every day for 365 days? What are the chances that it rains at least once in that year?

My math genius friend Josh came up with the answer for me:

Question #1:

There is a 70% chance it will not rain on Sunday. There is a 70% chance it will not rain on Monday. Therefore, the chance it will not rain on either day is (70/100) * (70/100), or 49%. So the chances it will rain on at least one day are 51%.

Question #2:

There is a 99% chance of no rain on any given day. Therefore, the chance it will not rain on 365 consecutive days is (99/100)^365, or 2.55%. The chances of at least one day of rain are 97.45%.

Basically, it’s:

probability it will happen in at least one of the periods = 1 – ((1 – probability) ^ periods)

This led into a discussion of the Monty Hall problem, which is strangely fascinating.