If he has an umbrella at both home and work, his chance of getting rained on while commuting is zero, because if it's raining he will have an umbrella handy.

Now, say it rains one day and he does NOT return his umbrella to work after taking it home with him to stay dry. From there, if it is raining and he is commuting home<->work, his chance is 50%, as one place has an umbrella and one doesn't, at a ratio of 1:1.

The same situation goes for commuting home<->[non-work]. Unless he borrows an umbrella.

So, I guess the logical answer would be 0+50+50/3 = 33% chance of getting rained on during a commute, AFTER his umbrella is moved from one place to another. If it's returned, or not yet moved, read on:

Say he does return the umbrella, or it hasn't been moved yet. He always has a 0% chance of getting wet while commuting home<->work. If he is traveling home<->[non-work], this remains at 50%. So, then it's 0+50/2, so 25% chance.

**EDIT** -

(4:23:24 AM) [x86] Ender: btw I don't really have time to read through the problem/solution that much, but just because of your conditions "if he does this, probability is this" and "if he does that, prob. is this"

(4:23:34 AM) [x86] Ender: i think it's wrong b/c of that, though i think you're thinking along the right lines

(4:24:04 AM) [x86] Joe: Yeah, that problem isn't all that great cause there's external factors.

(4:24:18 AM) [x86] Joe: For example, him getting wet is dependant on the probability it's raining when he's outside.

(4:24:36 AM) [x86] Joe: I just tried to figure out if he'll have an umbrella, should it be raining when he's outside.