Poll

1 vote (33.33%)
No votes (0%)
1 vote (33.33%)
1 vote (33.33%)
No votes (0%)

3 members have voted

Klopp
Klopp
Joined: Mar 10, 2020
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March 13th, 2020 at 9:24:02 AM permalink
A fair die is rolled until a 6 appears. My payout in dollars is equal to the number of rolls if no odd numbers were rolled and is zero otherwise. What is the conditional expected value of my payout given that no odd numbers were rolled until the appearance of a six ? You are told that I have played the game and have rolled a 6 without rolling odd numbers. What is the conditional probability that I got a payout of $1 in this situation?
Last edited by: Klopp on Mar 13, 2020
TigerWu
TigerWu
Joined: May 23, 2016
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March 13th, 2020 at 9:35:13 AM permalink
My gut instinct is that this sounds like a TERRIBLE game, but I don't know what the math would be so I could be wrong...haha
ThatDonGuy
ThatDonGuy
Joined: Jun 22, 2011
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March 13th, 2020 at 10:18:00 AM permalink
Is this a fair game? Not even close - the fair value is 37.5 cents.
The problem is not just that any odd number ends the game with a loss, but if you roll a 6 in either of your first two rolls, you lose money as well.
You get $1 if the first roll is 6; this has probability 1/6
You get $2 if the first roll is 2 or 4, and the second is 6; this has probability 1/3 x 1/6
You get $3 if the first two rolls are 2 or 4, and the second is 6; this has probability (1/3)2 x 1/6
and so on
The expected return is 1 x 1/6 + 2 x 1/3 x 1/6 + 3 x (1/3)2 x 1/6 + 4 x (1/3)3 x 1/6 + ...
= 1/6 x (1 + 2 x 1/3 + 3 x (1/3)2 + ...)
= 1/6 x (1 + 1/3 + (1/3)2 + (1/3)3 + ...)2
= 1/6 x (1 / (1 - 1/3))2
= 1/6 x 9/4 = 3/8

If you rolled a 6 without any odd numbers, the only way to win $1 is if your first roll was a 6. Just in case this is a homework problem, I'll leave the rest to you; just figure out the probability of winning $2, $3, $4, and so on. (Hint: what numbers do you need to roll to win $3?)
Klopp
Klopp
Joined: Mar 10, 2020
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March 13th, 2020 at 10:59:10 AM permalink
I don't agree about your answer for the expected payout. Also, the probability I asked about is a CONDITIONAL probability: the probability that the first roll was a six given that no odd numbers were rolled until the appearance of a six.
ThatDonGuy
ThatDonGuy
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March 13th, 2020 at 11:55:07 AM permalink
Quote: Klopp

I don't agree about your answer for the expected payout. Also, the probability I asked about is a CONDITIONAL probability: the probability that the first roll was a six given that no odd numbers were rolled until the appearance of a six.


What do you get for the expected payout, and how did you get that value?

And I understand that the probability was conditional; it's (the probability of winning $1) divided by (the probability of winning $1 + the probability of winning $2 + the probability of winning $3 + ...)
Wizard
Administrator
Wizard
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March 13th, 2020 at 12:03:33 PM permalink
Here is my interpretation of what the player gets for his $3 to play:

The player rolls a die until any odd number or a six is rolled. If that game ending roll is a 1, 3, or 5, then the players nothing. If that game ending roll is a 6, then the player will win $1 for every roll of the die (including the last roll of a 6).

If this is true, I get an expected win of 37.5 cents. There is a 75% chance of winning nothing. If the player does win, his average number of rolls is 1.5. Thus, the expected win is 0.25 * 1.5 = $0.375 .

In Klopp's last post he mentioned a conditional probability, which seems to indicate there is a misunderstanding here.
It's not whether you win or lose; it's whether or not you had a good bet.
IndyJeffrey
IndyJeffrey
Joined: Feb 10, 2013
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March 13th, 2020 at 12:30:47 PM permalink
There is only one situation where you get $1:

Roll a 6

Probability = 16.67%

Otherwise, your payout is not $1.

Unless the question is seeking a payout of $4 ($4-$3 ante), then there are more situations. One being:
Roll a 2 + 2 + 2 + 6

So, .33 * .33 * .33 * .17

Probability = 0.62%
Klopp
Klopp
Joined: Mar 10, 2020
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March 13th, 2020 at 12:35:31 PM permalink
The Wizard is right as usual. My formulation was not okay. I have repaired the formulation. My apologies to ThatDonGuy

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