## Interview Question

Analysts**Team:**Risk analysis

**Country:**United States

**Interview Type:**Phone Interview

@Anon -- The company is startup on credit card payment.

Here is my answer:

S1: Stop after 1st roll and get rewarded: expected reward = (1+2+3+4+5+6)/6 = $3.5

S2: Try 2nd time if

a) the first roll is less then 6: (1/6)*6 + (5/6)*(3.5)

b) the first roll is less then 5: (1/6)*6 + (1/6)*5 + (4/6)*(3.5)

c) the first roll is less than 4: (1/6)*6 + (1/6)*5 + (1/6)*4 + (3/6)*(3.5)

d) the first roll is less than 3: (1/6)*6 + (1/6)*5 + (1/6)*4 + (1/6)*3 + (2/6)*(3.5)

...

Hence c) is the stragety to maximize the expected reward.

Comments?

If we have an option to refuse a result and try again then why not wait till you get 6??? :P

The expectation is 3.5 either way.

On the first roll if you roll less than 3.5 (i.e 1-3) you should roll again. If you roll higher you would stay.

The probability outcomes of the second roll are the same as the first. So expectation here is 3.5. Once you decide to roll again you completely forget what the first dice gave you.

Define X = face value on dice upon rolling the dice the first attempt

Define y = threshold (to be determined). If X > y, then we stop after first die roll. If X <= y, we roll the dice once more. It is clear that 1<=y<=6

Define the average pay off to equal Z.

Then Z = { 3.5, X <= y,

[(y +1) + ... 6]/(6 -y), X > y

}

We now compute the expected payoff Z for different values of y

y = 1: Z = 3.5 x (1/6) + (2+3+4+5+6)/5 x (5/6) = 47/12

y = 2: Z = 3.5 x(2/6) + (3+4+5+6)/4 x (4/6) = 50/12

y = 3: Z = 3.5 x (3/6)+ (4+5+6)/3 x (3/6)= 51/12

y = 4: Z = 3.5 x (4/6) + (5+6)/2 x (2/6) = 50/12

y = 5: Z = 3.5 x (5/6) + 6 x 1/6 = 47/12

y = 6: Z = 3.5 x (6/6) = 42/12

Based on above, optimal strategy is to choose threshold y = 3. Expected payoff = $51/12 .

Is there just one die? If so, the expected reward of a roll is $3.5. So you should reject the first roll if it's less than that (if it's less than or equal to 3).

- eugene.yarovoi April 24, 2012The expected reward is: rewards from first roll + rewards from second roll. First roll = 1/6 * 6 + 1/6 * 5 + 1/6 * 4 = 2.5. Second roll = chance of getting that far * expected value of a roll = 1/2 * (3.5) = 1.75. So the total expected value of this game is 4.25.