Can anyone answer this probability question?

[rmani]

I had an interview today with an options trading firm, which is what i really want to go into after I graduate from college in a month. Anyway they asked me this applied probability question and I was a little stumped until the interviewer gave me a hint. Ok well here's the question:

You can play a 1 on 1 game against Jordan and have a 50% chance of winning 1 game. Should you agree to play him in a best of 1, 3 or 5 game series?


So i was a liittle stumped until she said what's 20 divided by 3. So from there i was able to work out 20/3 = 6.5 (approx) so you have a 65% chance if winning if you play a 3 game series, and a 40% chance if you play a 5 game series. The only thing I don't understand is where did she get the number 20 to use as a starting point? I don't understand how I could have solved that problem without her giving me the hint.

 

[SilverOne]

50% chance = 0.5 probability.

1 game => 0.5 probability => 50%

3 game => 0.5x0.5x0.5 = 0.125 => 12.5%

5 game => 0.5x0.5x0.5x0.5x0.5 = 0.03125 => 3%

I would say play 1 game, as you have the highest probability.

 

[TigerNSX]

Your question is a bit unclear.

Could you clarify?

 

[Teej]

If you have a 50% chance of winning playing just 1 game.

And you have an option to play 1,3, or 5 games....

If you win 1 out of 5 games...then you would have won 20% of the games. 20% probability in winning 1 of 5 games.

just a guess...kinda makes sense...

Or how about..... If your chances are even at 50/50 during a 1 on 1 game (the name Jordan is there to throw you off).

Then if you play 5 games, chances are you both might win 2 games, leaving 1 game up in the air. That 1 game is 20% of 5. So in order to win 3 games (60%), instead of 50% at two games, factor in that last 20 divided by 3 for the best chance. I think.....

I dont know.....as long as there is an odd number of games, your chances will always be greater in losing provided equivalent talent. 66% to win/lose 2 of 3, and 60% to win/lose 3 of 5.

You would be better off playing 1 game with a 50% chance. Its like flipping a coin (heads/tails) (true/false) (yes/no) (right/left) etc....

 

[jupiterfish]

depends on how much we were $$$ betting

 

[NetViper]

i would say it is just like flipping a coin... i dont think your probabilty increases at all if you play more games.

 

[Teej]

Yep Netviper, thats kinda where I was going. Sure, your chances increase in winning, but also in losing.

 

[Ojas]

Number of games is irrelelvant if it's decided ahead of time and you have the same chance of winning as your opponent.

However if you played 1 game and lose and MJ, being a good sports, asks if you want to make it best of 3 or 5. You should, of course, choose the higher number: 5. Having already lost the first game, you have a 25% chance of winning the remaining 2 games in a best-of-3, and 31.125% chance of winning 3 of the remaining 4 in a best-of-5.

Also, it gets interesting if you have to decide the number of games before you start, but you have a different level of skill than your opponent.

If you are more skilled than your opponent, you are better off choosing a bigger number of games. Conversely, if you are less skilled than your opponent, you are better off choosing a smaller number of games.

 

[PHOEN$X]

It's a trick question. Obviously your oponent isn't Michael Jordan if you both have the same odds of winning. That's just thrown in there to distract you.

I was thinking it's like a coin toss probability study as well. Here's an interesting simulation:

http://shazam.econ.ubc.ca/flip/

The outcome is more or less random.

 

[rmani]

hey guys, when I was originally told the question I proceeded as many of you have beeen doing. I used the coin-toss analogy figuring each game is independent of the other's outcome, and so I also thought my best odds of winning were to play 1 game. But for whatever reason she's like well here's a hint, what's 20 divided by 3 and we kinda worked from there. I didn't want to sound like I was totally lost so I refrained from asking where 20 came from. I have a feeling she messed up in telling me the problem, as she did make a couple of errors when trying to give me hints as to how to solve it. She at first also neglected to tell my the odds of winning even 1 game until I asked her. o well I'm just glad everyone is thinking along the lines that I was and I'm not a complete fool.

 

[Black Knight]

you mentioned jordan....michael is it?

chances of winning a game is 0%

but if the given is already a 1/2 chance of winning, then it's a different story.

first of all, i don't think the lady's answer is correct.

20/3 = 6.66.... = 66.66....%
20/5 = 4 = 40%

after going through some logic, my final answer is he'll have 50% chance of winning no matter how many games he played if the chance of each game is 50% win.

here's the logic (digital logic as well) 1=win, 0=lose

3 games - 8 possibilities of outcome (2 to the power of 3)

111 <- win
110 <- win
101 <- win
100
011 <- win
010
001
000

to win, he needs at least two wins and 4 of those possibilities satisfy the requirement

similarly, for 5 games, there are 32 possible outcomes (2 to the power of 5)

if you plot out the same list of 1s and 0s, you'll find 16 possible outcomes for at least 3 games won.

furthermore, if you're given 50% of winning, that means jordan also gets 50%. he has the exact same chance as you for 1, 3 or 5 games played. why would you have a 65% chance of winning in 3 games? he also played 3 games and only has 35% chance of winning? what if the question was asked of jordan? he then gets 65% while you get 35%?

make sense? if not, then i'll have to kill my digital logic design professor.....

 

[8000RPM]

Blue - you are right on!

That interviewer does not know what she is talking about.

I'd fire her if she worked for me.

 

[PHOEN$X]

That interviewer does not know what she is talking about.

I'd fire her if she worked for me.

Hold on. What if she was hot?

 

[Black Knight]

Hold on. What if she was hot?

then i'll ask her what are the chances of me scoring with her out of 1, 3 and 5 times.....

 

[8000RPM]

Hold on. What if she was hot?

I will just have to re-hire her as the receptionist for my company...

 

[Zuerst]

Mmm... I don't get her logic... I mean I think no matter how many games you play, the outcome is still 50-50.

I think of it this way. If you flip a coin, no matter how many times you flip it, the number of heads and tails are always going to be about equal. I never heard of flipping a coin 3 times would yield 66.6666666666% heads, and 40% heads if you flip it 5 times.

 

[Ojas]

Mmm... I don't get her logic...

Maybe her "hint" was just part of the trick question or just a joke they like to play on interviewees.

However, if she was serious about the hint, rmani maybe you should start looking for another company to work for. Then again, maybe "creative math" comes in handy in that industry.

 

[maomaonsx]

this seems like a pretty simpl binomial distribution problem (binary experienment repeated n times). I think she needs to get fired.

 

[rmani]

hey blue i think you're correct in the whole basic logic of the question. Yes it is MJ bubt iif we both have a 50% chance why would my odds improve iif we played more games? Doesn't really make sense. Since this was only the preliminary rounds of interviewing the woman who interviewed me was a human resources lady, she doesn't actually do any trading thus I agree she must have screwed up the question that she asked me.

Also Phoen$x I don't know if she was hot because the 1st rounds are just a telephone interview, hopefully if I make 2nd rounds they'll fly me out to interview with actual traders in person. I must admit though she sounded pretty cute.

Thanks for the replies fellas.

 

[Hrant]

50% chance = 0.5 probability.

1 game => 0.5 probability => 50%

3 game => 0.5x0.5x0.5 = 0.125 => 12.5%

5 game => 0.5x0.5x0.5x0.5x0.5 = 0.03125 => 3%

I would say play 1 game, as you have the highest probability.

Close but no cigar as the question is winning out of playing 1 gamer, out of 3 games or out 5 games.

Your analysis is correct if the question was winning all 3 or 5 games.

To win out of 3 games, you need to win 2 games. Probbability of winning two games is 25% (0.5x0.5). Similarly probability of winning 3 out of 5 games is 12.5% (0.5x0.5x0.5).

Therefore your best bet is to play him one game where your odds are highest at 50%.

Now she could have tossed you a curve by saying what if the requirement was winning all games consecutively, then what would have been your answer?

rmani, I think you are correct, either she was confused or she was trying to trick you .....

as fo all others who believe it doesn't matter how many games you play as the odds are the same, please come to my casino, do I have a game for you ......

 

[Ojas]

Close but no cigar as the question is winning out of playing 1 gamer, out of 3 games or out 5 games.

Your analysis is correct if the question was winning all 3 or 5 games.

Yup.

To win out of 3 games, you need to win 2 games. Probbability of winning two games is 25% (0.5x0.5). Similarly probability of winning 3 out of 5 games is 12.5% (0.5x0.5x0.5).

Therefore your best bet is to play him one game where your odds are highest at 50%.

Wrong!!! Duh! You are you considering winning a series in a sweep where MJ does not record a win! Add those cases into the picture and you're right back at 50%.

as fo all others who believe it doesn't matter how many games you play as the odds are the same, please come to my casino, do I have a game for you ......

Okay, I'll be MJ with my math you can be you with yours

Seriously, it's still 50%. In fact, it could be no other way, since from MJ's point of view, he's playing the same game as you!!!

Blue Knight's illustration explained all the outcomes of a best-of-three series. For illustration, here are the outcomes (W=win, L=lose)

LL = 25%
LWL = 12.5%
LWW = 12.5%
WLL = 12.5%
WLW = 12.5%
WW = 25%

Add up the numbers and you'll see that you have a 50% chance of winning, you have a 50% chance of losing (MJ has the same chances).

The same extends to best-of-5, 7, etc. series.

 

[Sig]

With the information she gave you in the format you presented the answer would be 1.

That said, I believe the interviewer presented the situation in a falty manner, assuming you presented her words exactly.

The Jordan comment is in fact a distraction.

The question:
*********************
You can play a 1 on 1 game against Jordan and have a 50% chance of winning 1 game. Should you agree to play him in a best of 1, 3 or 5 game series?
**********************

Here the main issue I see:
1) You only know that you have a 50% chance of winning 1 game; but you do not know the denominator of the equation.

For example, it is possible that you could have a 50% chance of winning 1 game..... but only if you play 10 games total. This obviously sways the per game odds well below 50% on a per game basis.

 

[cxr344]

Sorry, but your analysis is also totally incorrect. You've only calculated the odds of winning the first two or three games consecutively in a three or five game series. This ISN'T the only way to win however. Look at Blue Knight's post for a pretty detailed explanation. In the end, the odds are most definitely 50/50.

Don't feel too bad, it kinda stumped me for a minute too, and my initial guess was the same as yours (to pick the one game series). Intuively it makes sense, but just because the fact that you're playing against Michael Jordan gets stuck in your head. But thinking it through a little more, I had to ask myself "Why the hell would his 50% chance at winning each individual game be any different than my 50% chance?". Looking at it this way, its pretty much common sense.

Good trick question though, definitely.

To win out of 3 games, you need to win 2 games. Probbability of winning two games is 25% (0.5x0.5). Similarly probability of winning 3 out of 5 games is 12.5% (0.5x0.5x0.5).

Therefore your best bet is to play him one game where your odds are highest at 50%.

 

[PoohBEAR]

Your probability answer should of been.....Niether. An average Joe wouldn't be able to win Jordan. As a option trader you're dealing with reality probabilities which require the art of estimating, proximate, and a little research to back up your call or put. The reality of this question would be to ask her back that you would not risk this bet and loose your shirt. From all indication, Jordan is an expert when it comes to playing basketball.....an average Joe would win? I doubt it. The same goes to real life as you woun't want to loose your non-descritionary account when acting on something that you think--sometime the number lies. Just my 2cents.

 

[Hrant]

Yup, I fell for the same mistake !!

But, if I was evaluating this, the answer I would be looking for is still one game ......... why would you want to play more than one game if the odds are the same ..... unless you just like to play BB and/or with MJ ........