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 07-21-2012, 11:29 PM #2 Carpal \'Tunnel     Join Date: Mar 2010 Location: Philadelphia Posts: 6,011 Re: Probability Interview Question with an Asset Manager Umm.. Just a guess, but if you win 9 dollars if you bet, but win 10 dollars if you get to the final marble, my guess would be that you have to have odds overcome the difference between the profit of 9 dollars vs 10 dollars, so your odds would have to be a better expected value to win 9 dollars than 50/50 to win 10. Or am I completely wrong?
 07-21-2012, 11:39 PM #3 centurion   Join Date: Feb 2010 Posts: 104 Re: Probability Interview Question with an Asset Manager I'm not sure I'm interpreting the question correctly, but I think the answer is that it doesn't matter what you do. Your expected earnings is \$5. This is because every strategy is equivalent to picking the last marble, which has a 50% chance of being red a priori.
 07-21-2012, 11:44 PM #4 Carpal \'Tunnel     Join Date: Mar 2010 Location: Philadelphia Posts: 6,011 Re: Probability Interview Question with an Asset Manager But what if 10 red and 10 blues are taken out, (down to 80 marbles), then 5 blues get taken out in a row? The percentages have now changed, and when you pick you only get paid 9, but if you don't play, you have a chance to win 10. Like you said, I'm not sure if I'm interpreting this correctly (the post is a bit hard to follow in all honesty)
 07-22-2012, 03:30 AM #5 Carpal \'Tunnel     Join Date: Sep 2002 Posts: 8,896 Re: Probability Interview Question with an Asset Manager Nice problem. The best strategy is to never bet. In that case your EV is \$5. All other strategies have an EV less than \$5. You might think you can do better if you bet when the percentage of red marbles is greater than 60% since when that condition arises your EV would be greater than \$5. However, your EV would be even better by not betting because the last marble has the same probability of being red as the next marble, but you get paid more by not betting. As an extreme example, suppose all of the remaining marbles were red. You still shouldn't bet because you will only make \$9 instead of \$10 by waiting until the end. Last edited by BruceZ; 07-22-2012 at 05:56 AM.
07-22-2012, 03:40 AM   #6
Carpal \'Tunnel

Join Date: Sep 2002
Posts: 8,896
Re: Probability Interview Question with an Asset Manager

Quote:
 Originally Posted by randomwalk I'm not sure I'm interpreting the question correctly, but I think the answer is that it doesn't matter what you do. Your expected earnings is \$5. This is because every strategy is equivalent to picking the last marble, which has a 50% chance of being red a priori.
They aren't equivalent. If I bet on the first draw, my EV is \$4. Your expected earnings are \$5 only if your strategy is to never bet. Any other strategy has an EV less than \$5.

Note that if they only paid you \$9 when the last marble is red and you never bet, then you could do equally well by betting when all of the remaining marbles are red, but betting any other time would do worse. If they paid you an amount less than \$8.98, there there could be a time when it becomes better to bet.

Last edited by BruceZ; 07-22-2012 at 07:10 AM.

07-22-2012, 03:40 AM   #7
Carpal \'Tunnel

Join Date: Mar 2010
Posts: 6,011
Re: Probability Interview Question with an Asset Manager

Quote:
 Originally Posted by BruceZ As an extreme example, suppose all of the remaining marbles were red. You still shouldn't bet because you will only make \$9 instead of \$10 by waiting until the end.
Sigh. Well done Bruce. I thought I had it with my logic. I wonder if they'd think I was an idiot with my answer.

07-22-2012, 03:58 AM   #8
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Join Date: Sep 2002
Posts: 8,896
Re: Probability Interview Question with an Asset Manager

Quote:
 Originally Posted by wil318466 Sigh. Well done Bruce. I thought I had it with my logic. I wonder if they'd think I was an idiot with my answer.
I would be surprised if anyone got that right under the pressure of a phone interview. If they did, they should probably be hired on the spot.

 07-22-2012, 04:07 AM #9 Carpal \'Tunnel     Join Date: Mar 2010 Location: Philadelphia Posts: 6,011 Re: Probability Interview Question with an Asset Manager I'm having an issue with the EV though. If the blue to red ratio is above .57, it'll create a 5.13 EV (9 dollars multiplied by .57 probability). That's obviously higher than the 10*.5=5 EV... so why did i miss your logic about not betting when almost all red (say, 9*.9=8.1 EV)? I dunno.. I'm missing something obvious here.
07-22-2012, 04:14 AM   #10
Carpal \'Tunnel

Join Date: Sep 2002
Posts: 8,896
Re: Probability Interview Question with an Asset Manager

Quote:
 Originally Posted by wil318466 I'm having an issue with the EV though. If the blue to red ratio is above .57, it'll create a 5.13 EV (9 dollars multiplied by .57 probability). That's obviously higher than the 10*.5=5 EV... so why did i miss your logic about not betting when almost all red (say, 9*.9=8.1 EV)? I dunno.. I'm missing something obvious here.
If your EV is 5.13, you don't compare that to 5, you compare it to the EV that you would get if you wait until the end. The last marble has the same probability of being red as the next marble, as do all the remaining marbles. If you had to place a bet, the last marble would have the same EV of 5.13 as the next marble. But the last one pays better when you don't bet, so it has a higher EV.

It's like in blackjack when the count goes high. On average the count will be the same on the next hand that it is on the last hand of the shoe.

Last edited by BruceZ; 07-22-2012 at 05:29 AM.

 07-22-2012, 04:17 AM #11 Carpal \'Tunnel     Join Date: Mar 2010 Location: Philadelphia Posts: 6,011 Re: Probability Interview Question with an Asset Manager Ok, so if there were 10 marbles left, 9 red 1 blue, its .9. So .9 * \$9 = 8.1, much higher than 5, which is .5 * \$10, but we shouldn't be comparing 8.1 to 5, it should be 8.1 compared to .9 * \$10, which is 9. Got it. I'm dumb.
07-22-2012, 04:38 AM   #13
journeyman

Join Date: Sep 2004
Posts: 360
Re: Probability Interview Question with an Asset Manager

Quote:
 Originally Posted by wil318466 Ok, so if there were 10 marbles left, 9 red 1 blue, its .9. So .9 * \$9 = 8.1, much higher than 5, which is .5 * \$10, but we shouldn't be comparing 8.1 to 5, it should be 8.1 compared to .9 * \$10, which is 9. Got it. I'm dumb.
0.9 * 9 - 0.1 * 1 < 0.9 * 10 - 0.1 * 0

I wrote this out and saw I would never decide to bet.

No reds: 0 * 9 - 1 * 1 < 0 * 10 - 1 * 0

 07-22-2012, 05:33 AM #14 Carpal \'Tunnel     Join Date: Sep 2002 Posts: 8,896 Re: Probability Interview Question with an Asset Manager Also, the EV of betting isn't 5.13 in your example where the probability of red is .57. It would be .57*9 + (1-.57)*(-1) = \$4.70 You need the probability of red to be greater than 60% for the EV to become greater than 5. But that doesn't matter because the probability that the last marble is red would be the same.
07-22-2012, 08:10 AM   #15
veteran

Join Date: Nov 2009
Posts: 2,697
Re: Probability Interview Question with an Asset Manager

Quote:
 Originally Posted by BruceZ As an extreme example, suppose all of the remaining marbles were red. You still shouldn't bet because you will only make \$9 instead of \$10 by waiting until the end.
Don't you get to keep betting if you win? This seems to imply that you do.
Quote:
 Originally Posted by mburke05 At any point you can bet 1\$, if it hits red you win you win 9\$ plus your 1\$ back. If it hits blue you lose and are no longer allowed to wager because you're broke.

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