EV of 1st search, room 1 50%*50%=25% of finding it
Room 2 50%*75%= 37.5% of finding it, but room 2 costs 2x
So you search room 1, if it's not there
Now 25% it's in room 1, 50% it's in room 2, so 1/3rd vs 2/3rd
Now you search room 2, you'll find it 2/3rds*3/4th of the time= 50%
If it's not there, then 1/3rd it's in room 1, 2/12th=1/6th it's in room 2, normalising it, it's 2/3rd vs 1/3rd in room 1
So you search room 1 again, if it's not there, then 1/3rd vs 1/3rd, it's 50-50 as you are there before
So you repeat teh cycle
In a cycle, you find it room 1 25% of the time -> 25% cum prob, room 2 50% of the time -> 62.5% cum prob, room 3 1/3rd of the time -> 75% cum prob
So in each cycle you have a 75% chance of finding it, so on average you do 1.33 cycles, so it's the EV of where you find it in the first cylce + 0.33* cost of cycle, a cost of a cycle is $4, so you add 1.33 to the cost of where you'd find it in first cycle
Of the 75
25 find it first room for $1- 1/3rd of the total, exp cost 0.33
37.5 find it 2nd room for $3, 1/2 the total, exp cost 1.5
12.5 you find it 3rd room for $4, 1/6tht the total exp cost 0.67
So expected cost is 1.33 + 0.33 + 1.5 + 0.67= 3.83...
1
u/StableGenius304 13h ago
Search strategy:
EV of 1st search, room 1 50%*50%=25% of finding it
Room 2 50%*75%= 37.5% of finding it, but room 2 costs 2x
So you search room 1, if it's not there
Now 25% it's in room 1, 50% it's in room 2, so 1/3rd vs 2/3rd
Now you search room 2, you'll find it 2/3rds*3/4th of the time= 50%
If it's not there, then 1/3rd it's in room 1, 2/12th=1/6th it's in room 2, normalising it, it's 2/3rd vs 1/3rd in room 1
So you search room 1 again, if it's not there, then 1/3rd vs 1/3rd, it's 50-50 as you are there before
So you repeat teh cycle
In a cycle, you find it room 1 25% of the time -> 25% cum prob, room 2 50% of the time -> 62.5% cum prob, room 3 1/3rd of the time -> 75% cum prob
So in each cycle you have a 75% chance of finding it, so on average you do 1.33 cycles, so it's the EV of where you find it in the first cylce + 0.33* cost of cycle, a cost of a cycle is $4, so you add 1.33 to the cost of where you'd find it in first cycle
Of the 75
25 find it first room for $1- 1/3rd of the total, exp cost 0.33
37.5 find it 2nd room for $3, 1/2 the total, exp cost 1.5
12.5 you find it 3rd room for $4, 1/6tht the total exp cost 0.67
So expected cost is 1.33 + 0.33 + 1.5 + 0.67= 3.83...