1, i would bet that the next time is still head..the coin is probably biased.

1) Won't bet. Odds are high coin is biased, as above. the amount doesn't change to a lousy bet, unless you're talking odds. The amount only matter if the odds change, and if it affects you cos of your risk appetite.

2) Ask him to get out of line. If you got no guts to express your opinion here, forget about the markets.

1) You don't know what odds are being given so you can't make an assessment of the risk-reward.

2) You don't queue up for a premiere.

1) If we were talking a standard quarter (i.e. 50/50 chance of heads or tails) I would bet whatever amount you wanted regardless of previous results.

2) I would probably have to crush anyone who tried to cut me

1) The results of the next coin flip is an independent event. Its P(x=tails)=.5. Your advantage in knowing the previous flips is that over the long run the average result will mean revert to its true probability(basic MCMC). While you have no advantage on the next coin flip, you do have an advantage in saying that out of the next say 45 flips there will be more tails than heads.

2) obiously a fit question but my answer would: Ask him or her to go to the back of the line. If he refuses to, I would resist my first urge to beat the shit out of him (can't watch the premeire in jail) and I would probably publically embarass him by doing something like shout to the line "heeeyyyy everybody, this guy is sooooooooo important he doesn't have to wait in line. Let's give him a round of applause for giving us the opportunity to be in his presence."

1) I'd take the bet, while you have 50/50 chances of heads or tails either way, you go ahead and try to flip 6 straight heads thats like P(x=heads), with 6 successes out of 6 trials, =.0156

2)I'd let it go, if they wanted to see either aggressiveness or calm under pressure, I'd direct them to the "military" section of my resume. People never cut me in line, and if someone did they'd obviously be looking for a fight- which wouldn't be worth it.

"While you have no advantage on the next coin flip, you do have an advantage in saying that out of the next say 45 flips there will be more tails than heads."

This is false.

1) Prior events do not change an equal weighted probabily event's next occurence.

However, if there was a function that stated everytime x occurs the probability of y occuring increases by 10% I would take the bet because the probability of tails would now be 88.6%

2) I would never go to a premiere because no movie since I was 6 (Jurassic Park) has really gardnered that much of my interest. But IF I went and someone cut me I would be like hey the line is there and if they stayed. I would either tell someone (if I wanted to be a bitch), punch him (if I wanted to be macho), or let him stay there because he is probably trashy person.

the odds of getting 5 heads in a row is (1/2)^5= 3.125%, while not impossible, i would bet the coin is bias and would bet on heads.

i think they want to make sure you are aggressive and wont get run over on a trading floora, tell the guy to get the hell to the back of the line.

I can't believe the answers I'm seeing for #1. No you don't bet on it, period. It's not a trick question. There are not enough observations to conclude that it's a "biased coin." Come on guys.

The idea of mean reversion of the average over increased iterations is the basis of Monte Carlo Method, Metropolis-Hatings, ect.. If we use a payout of "0" for heads and "1" for tails Our sample shows that the "Expected Payout =P(X=tails)(Payout for tails=1)(Number of iterations)= 2.5. Our payout from the prior is zero. Over 50 iterations we would expect the payout to be 25 (with results being 25 heads and 25 losses). (Assuming zero variance from true mean....a big assumption, but we could use numerous variance reduction techniques). If at iteration 6 you bet 1 dollar of the next 45 throws, you can predict that will contain 25 tails and 20 heads, giving you a payoff of five dollars. The risk you incur is in the variance, but the variance could be reduced numerous ways (most easily by increasing the iterations or by different sampling methods such as flip two coins with a payout of two if both tails, zero if mixed, and minus two if heads.) If we put the variance in play, the variance in the sample probabilty of "tails" result where theta equals the true population mean is (theta)(1-theta)/N=.005....so for our E(Expected Payout) =(P(X=tails)which we now know is N~(.5,.005) over a sample of 50)(Payout for tails=1)(Number of iterations).

Bottom line: go long tails with a 50+ horizon

"If at iteration 6 you bet 1 dollar of the next 45 throws, you can predict that will contain 25 tails and 20 heads, giving you a payoff of five dollars."

What world are you living in? Heard of a random walk? Martingales? etc etc

1) I would not take the bet. The Expected value is 0 and certainly I could find something better to risk my money on. 5 heads isnt nearly enough to even consider that the coin is biased unless there was something else visually off about the coin. Coin tosses are independent events anybody who argues they had an edge b/c of past results dosent get it.

2) I would tell the truth: If the guy was way bigger then me and appeared likely to pound me into a bloody pulp I would let it go, if not I would nicely tell him to go to the end of the line.

2) I would tell the guy that he can have my spot in line for the right amount of money, otherwise go to the back of the line...i think thats what a trader mite be looking for

I think people are missing the point of question #2.

The only time you'd give a shit about someone cutting in line is when your spot in that line was for the last ticket. If you were further up in line you wouldn't give a shit since the movie will start when it starts. If you're far enough back in the line you don't care because you weren't going to get a ticket any way (and shouldn't have been waiting for a ticket in the first place). It reminds me of an interview question I got from Bain Capital, where the more intuitive response is only applicable on the margin.

...i just read tlips' message and dude you need to just think rather then post. The bet is on the next toss. The coin is 50/50. The fact that the 5 previous came up one way dosent matter...in fact it happens quite regularly...3.125% of the time. I cant believe a college graduate would argue to take that bet for any other reason then a belief that the coin was biased (although they have no reason to believe that based on the 5 observation sample).

i'd tell em that you don't take 50/50 bets, i'd say "i bet on sure things" and you bet with considerable better odds. if you want 50/50 action go to a casino, and previous events don't affect probabilities.

casinos dont give u 50/50 odds

Tlips you need to retake basic probability. All future throws are independent, period. There is no "favortism" going forward because obviously the coin doesn't "remember" how it came up before. The only reason it will revert to the mean is because the number of trials becomes large.

I'm going to play devil's advocate for a moment. Now, initially I would agree that you should not take the bet, since if it truly is a quarter then each toss is independent. Thus it would be a fair bet, which might as well be no bet at all. In fact, statistically speaking, it would be strange (over a macro-horizon of many trials) NOT to see "statistical impossibilities"; the chances of it coming up heads then tails every other throw is also low.

Now, the devil's advocate part is (and correct me if my statistics is off, it's been a while) that 5 throws in a row coming up heads is statistically significant at a 5% level; that is, you may reject the null hypothesis that the coin is fair. Then again, if you have no indication that the coin IS biased (that is, if the trader just pulled a coin out of his pocket and flipped it five times, no tricks up his sleeve or obvious incongruities) I'd probably want a higher threshold of significance.

Good thing these postings are anonymous... re question 1, future flips are independent, period. Tlips' posting is scary, because apparently he/she has seen some fairly advanced statistics but has no idea. Popping is correct, although also dropping fancy stats terms without understanding them. I don't get what martingales or random walks have to do with anything. (if the realizations of the coin flip were a martingale, then betting heads would be correct!) MCMC and Metropolis-Hastings? Tlips wants to appeal to a law of large numbers, but the (or a) LLN just says sample moments converge to population moments.

As a side note, most of the responses here showcase that, in the real world, everyone is a Bayesian. People have different priors about whether the coin is fair, and this affects their decisions about taking the bet. It's pointless to argue whether a 3.25% chance of 5 straight heads is "small enough" to make you decide the coin is biased. Depends on your prior.

Personally, I'd answer the question by saying it depends on what side of the bed I woke up on. Assuming a fair coin, I'm indifferent between taking a fair bet and not, for $10 anyway. Make it $1000 and I say no bet, because then risk aversion starts to have some bite.

Please review Law of Large Numbers because I think you are the one who doesn't know what he is talking about: http://en.wikipedia.org/wiki/Law_of_large_numbers

Do us a favor and show us how LLN implies convergence of second moments?

And while you are at at, pray tell what fancy stats terms I dropped?

"I don't get what martingales or random walks have to do with anything. (if the realizations of the coin flip were a martingale, then betting heads would be correct!)"

You can't see how the P&L from a string of bets on a game of coin flipping is a random walk? Or how it's a martingale?

Please explain how, if the P&L from this game is a martingale (which it is), betting heads would be expected to be profitable? That violates one of the conditions of a martingale!

I can no longer listen to most of this reasoning.!!!! Forgive me. As someone who has interviewed new recruits and Oxbridge candidates for many a year, I feel a need to intervene.

POINTERS

1. Very important to correctly "hear" the question.

2. Don't try to second guess..(as a trader or sales person, I need to sound like a man attitude)

Question number one is a Middle School math/probability question:

http://mathforum.org/library/drmath/view/60767.html

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Question number two:

In the UK and most European countries, when you attend a movie premier, you get an advanced ticket with assigned seating. A person cutting in will not make any difference to your seat assignment and likelihood of seeing the movie.

I do not know how a movie premier works in the USA. But don't you have a ticket reserved or in hand, before you stand in line to enter the movie theatre itself?

So it depends on what firm, interviewer's background and experience.

============================================

I noticed some posters had the correct thought process (maybe they are Brits!)

Would it be helpful if we had a sticky on such questions and answers?

I do not mind starting it off, if the board monitors approve.

Interesting, I doubt many people here in the States would have known that about number 2.

"(if the realizations of the coin flip were a martingale, then betting heads would be correct!)"

uhh..

Cheers devin--I had to google this question after your last post, couldn't get any work done just thinking about it. LOL

Apparently, movie premiere tickets are given out in advance. *** Key word = premiere ***

http://www.ehow.com/how_2072185_tickets-movie-premiere.html

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So how many of you will still beat up, humiliate, etc. the person who cuts in line at a movie premiere?

If you were an interviewer and someone answered the question this way--would you want to work with them as part of a group/ team?? Is this a proper fit for the bank?

I think not.

1) the coin is either biased, or if you know it's not, the past flip record is irrelevant.

2) I would show the person the back of the line. Even if there were assigned seats, still doesn't allow you to cut.

Morning Jimbo- I know you are a trader--- and probably the most refined, as I have commented before. But for #2, what if "the someone" at the premiere was a staring actor, royalty, president, disabled, etc.?? Still show them the back of the line??

hmmmm Don't disappoint me Jimbo.

Are you serious??? I am only a high school student (yes, I want to work in IB), but I was surprised by stupidity of some of the answers. THE PREVIOUS FLIPS ARE IRRELEVANT. Are the people who answered something else really i-bankers? How can they know so little about math? Btw, if you think something like this: "If at iteration 6 you bet 1 dollar of the next 45 throws, you can predict that will contain 25 tails and 20 heads", why do you not play roulette instead of working long hours? You could make a lot of money...

adammarek, I dont think it reflects well on you by trying to state your intellectual superiority on this board by critcizing other people and their answers, of a very basic probability question. It's like me going to your ap calculus high school class and laughing at their inability to quickly calculate definite integrals in their head. See what Im getting at? (Calculating that isnt very impressive.) be humble and dont talk/write like a jerk.

I am sorry if my post sounded offensive, i did not mean it that way. I am not trying to "state my intellectual superiority", because I do not think that I am intellectually superior in any way. It just seems to be ironic to me when I read people saying that i-banking is only for the best people from the very best universities and then "predicting" a coin toss.

"I am sorry if my post sounded offensive, i did not mean it that way. I am not trying to "state my intellectual superiority", because I do not think that I am intellectually superior in any way. It just seems to be ironic to me when I read people saying that i-banking is only for the best people from the very best universities and then "predicting" a coin toss"

don't sweat it dude....this type of basic probablility question is what I would characterize as "fundamental life skills" and people really should have a better handle on it.

If not, let's flip some coins and I'll take the other side, in size.

Well said Jimbo. You did not disappoint. I guess I should heed my own advice and read the question better--(cut in line). I probably would let it pass, maybe that is why I get a tight gut when I am on the trading floor.

A virtual coin flip.

http://www.betweenwaters.com/probab/flip/coinmainD.html

Good for the times you are avoiding the markets. Like I am now.

how about this one....roll of a die, or flip of a coin, whatever you want....heads you get paid 2x what you would have for the year, tails you work for free.

would you do it?

If the question clearly states that the coin in unbiased, of course we know that past trials are irrelevant.

Probably the more interesting part is the thought process where the interviewee goes on to assume that is not the case in a real world, where other factors come into play, e.g. with 5 consecutive heads, what is the probability that the coin in biased?, and then come to the conclusion that odds are good that it is not a 50/50 game, and definitely not stacked in your favour. So why take it?

read the black swan by taleb..he writes few pages about this problem...i bet you the interviewer read it also cheers luke

Adam & Jimbo: No- I agree this is basic enough to be life skills, but I myself have refrained from ridiculing people with much worse mistakes (relative to their exp/knowledge). Maybe its just cultural norms though (Im Japanese).

Adam: I don't think you should trust that the people on this board are representative of the talent on a BB trading floor. imo There's maybe only 7-9 others besides Jimbo.

way too much of a risk to bet your entire salary on a coin flip, shows poor risk taking decisions. dont wanna risk $100m on a "coin flip" trade.

say you make 500K, you're risking 50/50 to make either 1 million or 0. Your expected return would be 500K so there's really no point in taking additional risk when odds average out to you earning original salary.

"way too much of a risk to bet your entire salary on a coin flip, shows poor risk taking decisions. dont wanna risk $100m on a "coin flip" trade."

there's no right or wrong...depends on a number of things. it's generally asked to 1st year analysts btw...so that depends on family wealth, debt things like that.

I would take the 1 dollar bet. People are risk averse and usually try to not lose more than try to win. The probability of landing heads six times in a row is (.5)^6, but the 6th flip is its own event, thus keeping the probability of that individual toss at one half. I would not take the 10 dollar bet.