This can be done with a simple program on a graphing calculator. (I did not check your work up to this point, but 7% seems a bit high for the likelihood of getting struck by lightning twice - or maybe I've just been lucky.)
Okay so I have 1000 houses in an area where lightning strikes one house at random every week, and I need to approximate the probablity of lightning striking the same house twice (or more, presumably) within one year (52 weeks), and then for two years. I'm allowed to use the Stirling Formula: .
Now if I did this right, with n multichoose k = I have possible outcomes with favorible outcomes which expanding with and then dividing leaves me with (kind of messy). Is there some way to further simplify this before plugging it into Stirling, or perhaps a simpler way of going about this altogether?
If I'm reading this problem right, you shouldn't be using the formula for at all.
Picture yourself living in one of the 1000 houses. Every week, you have a .1% chance of getting hit, and a 99.9% chance of not getting hit. So your chances of not getting hit two weeks in a row is 99.8%. So after 52 weeks, you have a 94.93% chance of not getting hit, and a 5.07% chance of getting hit. Wouldn't then the chance of getting hit twice in that same timeframe be .26% ?