SOLUTION: I'm having trouble with this problem. I get parts of it but can't seem to understand what the total variable is. Here is the problem: A town has an ice cream shop, a bowling alley

Question 867943: I'm having trouble with this problem. I get parts of it but can't seem to understand what the total variable is. Here is the problem:
A town has an ice cream shop, a bowling alley, and a park. Students hang out at one of these 3 places on a Friday night. A given student re-evaluates the choice to be at a particular location every 10 minutes as follow:
1. If at the park, the student goes to the bowling alley 10% of the time, the ice cream shop 20% of the time, and the otherwise stays put.
2. If at the bowling alley, the student goes to the park 5% of the time, to the ice cream shop 10% of the time, and otherwise stays put.
3. If at the ice cream shop, the student goes to the park 35% of the time, to the bowling alley 15% of the time, and otherwise stays put.
The police drive by the park and tell the students there to go home. Estimate the fraction of students that was told to go home, (from the park).
I get what percentage of the time for each occasion the student goes to the park, but how can I make this into a formula so that I can solve it when each occasion is conditional and there is no clear total variable given. Any help with this will be greatly appreciated. Thank you very much.
Answer by richwmiller(17219) (Show Source): You can put this solution on YOUR website!
Let's start with 100 in each place
b(1)=100 i(1)=100 p(1)=100
1)b(2)=b(1)+.10*p(1)
1)i(2)=i(1)+.20*p(1)
1)p(2)=.70*p(1)
2)p(2)=p(2)+.05*b(1)
2)i(2)=i(2)+.10*b(1)
2)b(2)=.85*b(1)
3)p(2)=p(2)+.35*i(1)
3)b(2)=b(2)+.15*i(1)
3)i(2)=.50*i(1)

0)b(1)=100
2)b(2)=.85*b(1)
1)b(2)=b(1)+.10*p(1)
3)b(2)=b(2)+.15*i(1)
85+15+10=110
i's
0)i(1)=100
3)i(2)=.50*i(1)
1)i(2)=i(1)+.20*p(1)
2)i(2)=i(2)+.10*b(1)
50+20+10=80
p's
0)p(1)=100
1)p(2)=.70*p(1)
2)p(2)=p(2)+.05*b(1)
3)p(2)=p(2)+.35*i(1)
70+5+35=110
We are only interested in p
110 in the park after 10 minutes
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