Probability

Working forwards in time:
P x 1/2 x 1/2 x 1/2 = 1/64

Simplified this is:
P/8 = 1/64

Therefore P = 6/64 = 1/8 (ANSWER B)

Alternatively, we could work backwards in time from 1/64:
1/64 x 2 x 2 x 2 = P

Again P = 1/8

I believe it's {B}

let initial probability = x
1st day = x/2
2nd day = x/4
3rd day = x/8 -(1)

Given: probability after 3 days = 1/64 -(2)

equation (1) & (2)

x/8 = 1/64

x = 1/8

Answer [spoiler]{B}[/spoiler]

AbhiS 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 of finding it has dropped to 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

[spoiler]
OA - D OR B, I have 2 sources who have sighted 2 different OA for the same question.
However I feel the OA should be D

This is how i reached to the D which i feel is correct because it mentions that - Each day after an item is lost the probability of finding that item is halved

i used the formula - (x)*(1/2*x)*(1/4*x) and solved; simplified solution (1/2)*(1/2*1/2)*(1/2*1/2*1/2) = 1/64

The other explanation was -

x * 1/2*1/2*1/2 = 1/64
Hence x = 1/8[/spoiler]

The second method was correct (x = 1/8)
The reason the other method is wrong is that the formula has "x" repeated:
(x)*(1/2*x)*(1/4*x) = 1/64
The probability is not multiplied by half the probability; it is only multiplied by half.
So the formula should just be:
(x)*(1/2)*(1/2)*(1/2) = 1/64
(1/8)*(1/2)*(1/2)*(1/2) = 1/64
Therefore x = 1/8


The error of also multiplying by "x" each day is akin to saying
P(Day 0) and P(Day 1) and P(Day 2) and P(Day 3) = 1/64
which has no justification in this context.

In other words:
Day 0: P = 1/8
Day 1: P1 = 1/8 x 1/2 = 1/16
Day 2: P2 = 1/8 x 1/2 x 1/2 = 1/32
Day 3: P3 = 1/8 x 1/2 x 1/2 x 1/2 = 1/64

A shorthand formula would use powers (indices), where "^" denotes to the power of.
(Starting probability) x (fractional daily change)^(no. of days) = (final probability)

P x (1/2)^3 = 1/64

So, P/8 = 1/64 -> P = 1/8.

This formula is the principle method of calculating compound interest on money invested, for example:

(Starting Money) x (100% + annual interest rate%)^(Number of years) = (Final sum)

Note that care must be taken when using such formulae; understand whether the scenario involves starting the interest calculations on year 0 or year 1, and adjust accordingly.

AbhiS 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 of finding it has dropped to 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

We can PLUG IN THE ANSWERS, which represent the initial probability.
Each day, the probability is multiplied by 1/2.
The correct answer choice will yield P = 1/64 on the 3rd day.
Answer choice A is way too small, since it implies that P = 1/2 * 1/32 = 1/64 after only one day.
Eliminate A.

Answer choice C: 1/4
1st day: P = 1/2 * 1/4 = 1/8.
2nd day: P = 1/2 * 1/8 = 1/16.
3rd day: P = 1/2 * 1/16 = 1/32.
The resulting probability is too great, implying that the correct answer choice must be SMALLER.

The correct answer is B.