Had this in an interview today and I think I boggled it up pretty bad, was at the end of a 1.5 hr interview that consisted of a 45 min options problem before this one, and basically nerves got the best of me.

This was on the phone so a few of these assumptions were unfortunately not articulated very well (by him in my opinion).

The guy also called me 2 hrs later than our call was supposed to be so maybe I'm just more frayed than normal and this is stupid ezpz, but over the phone this really stressed me out; was hoping to hear what you guys thought might be an efficient way to solve this.

Question is as follows:

This is a gambling game.

There is a jar with 100 marbles, 50 red and 50 blue.

One marble is taken from the jar at a time, without replacement. You can see what color that marble is.

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.

((OP Note: The interviewer said 'if you win', and didn't specify that you could not choose red or blue, so I made a derived decision tree based on this and I agonized pretty embarrassingly about this for about 15 minutes before I corrected myself))

However, you can also choose to not bet and not pay your 1$. If you do not bet, and the final marble drawn is red, you win 10$, if it is blue, you win 0$.

How do you play the game? (Assuming your intention is to maximize profit).

I'll give a spoiler with half the answer eventually (I got half before he told me "In an effort to save time, I'm going to end it there", you can tell the interview went pretty miserably for me :/). But I'm 99% sure after the "We're going to have a intra-firm meeting. We will be in touch if there are next steps for you by the end of this week." I was hosed.

He didn't give the other half, but I was curious, for closure's sake.

Thanks guys!

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?

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.

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)

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.

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.

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.

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.

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.

If people want some more, here's what my friend sent me from a hedge fund interview he just had. He mentioned his boss made one girl cry in an interview because she couldn't answer some questions (not these, just in general). These aren't too difficult, but kinda fun if you aren't in a pressure situation. It's always easier to answer these sitting at home and you have time to think about it.

"Dude. It was like 5 rounds. They made me compute ytd returns and CAgR in my head on the spot. I had a take home case study that I had to present to like 5 ppl over the course of three hours

there were a couple. one question was, if you have 10% gains for each month the first 6 months of the year and then 10% losses each month for the second half of the year, are you net up or down for the year? What is the YTD return? What if you have that same scenario and someone subscribes in an equal amount of capital after the first 6 months that you started with in the beginning of month 1? Are you up or down for the year? what is the YTD return?

if you start with 20 dollars in capital and 5 years later come out with 70 dollars of capital, what is the CAGR (compound annual growth rate)?"

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

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.

Don't you get to keep betting if you win? This seems to imply that you do.