to halve or not to halve

shahdevine wrote:Each day after an item is lost the probability of finding that item is halved. If 3 days after a certain item is lost the probability 1/64, what was the initial probability of finding the item?

a)1/32
b)1/8
c)1/4
d)1/2
e)1

oa is d...plz show set up! thx

Normally I don't reply to posts in the wrong sub-forum (this is the DS forum, not the PS forum!), but it's a fun question, so I'll make an exception.

It's also a great illustration of when backsolving is a great strategy. We know that the correct answer is one of the choices... we can just multiply the choices by 1/2 3 times until we find a match.

When we backsolve, we want to start with either (B) or (D); doing so guarantees that, as long as we can figure out which direction to go, we won't have to check more than 2 choices.

If we start with (B), we get:

starting prob: 1/8
after day 1: 1/8(1/2) = 1/16
after day 2: 1/16(1/2) = 1/32
after day 3: 1/32(1/2) = 1/64

Does 1/64 match the info in the question? YES! Therefore, (b) is the correct answer (contrary to the original post).

If we had gotten a result greater than 1/64, we'd know that we need to start with a smaller number; if we had gotten a result less than 1/64, we'd know that we need to start with a bigger number.

We can, of course, also do this question algebraically.

If we let the original probability be x, we get:

x(1/2)(1/2)(1/2) = 1/64

x(1/8) = 1/64
x = (1/64)(8/1) = 8/64 = 1/8... choose (B).