Thanks Brent. I edited my post to add the information.

p(1/2)^3 = 1/64
p = 1/8

The OA = D
I feel it is wrong.

The explanation:-

The probability of finding the item on the first day can be expressed as x/y

The probability of finding the item on the second day can be expressed as x/y * 1/2

The probability of finding the item on the third day can be expressed as x/2y * 1/2

The probability of finding the item after 3 days: x/y * x/2y * x/4y = x^3/8y^3 = 1/64 => x/y = 1/2


Experts please help!

Here are some flaws that I see in the explanation:

The probability of finding the item on the first day can be expressed as x/y
This is the probability at the start of day 1, because we know at the end of the day the probability is halved

The probability of finding the item on the second day can be expressed as x/y * 1/2
This is the probability at the start of day 2

The probability of finding the item on the third day can be expressed as x/2y * 1/2
This is the probability at the start of day 3

The probability of finding the item after 3 days: x/y * x/2y * x/4y
This is incorrect.
This is saying the probabilty after 3 days = probability at the start of day 1 x probability at the start of day 2 x probability at the start of day 3

Not sure I'm following the OA myself.

Suppose the original probability is p. That means that on the morning of Day 1, after ZERO days of searching, our probability of finding the item is p.

On the morning of Day 2, we haven't found the item, so after ONE day of searching, the probability that we'll find the item today is now (1/2)p.

On the morning of Day 3, we STILL haven't found the item, so the probability is (1/4)p.

On the morning of Day 4, three days have elapsed, and we still haven't found the item, so the probability is (1/8)p.

Given that (1/8)p = 1/64, we'd have p = 1/8.