## Welcome to BrainDen.com - Brain Teasers Forum

 Welcome to BrainDen.com - Brain Teasers Forum. Like most online communities you must register to post in our community, but don't worry this is a simple free process. To be a part of BrainDen Forums you may create a new account or sign in if you already have an account. As a member you could start new topics, reply to others, subscribe to topics/forums to get automatic updates, get your own profile and make new friends. Of course, you can also enjoy our collection of amazing optical illusions and cool math games. If you like our site, you may support us by simply clicking Google "+1" or Facebook "Like" buttons at the top. If you have a website, we would appreciate a little link to BrainDen. Thanks and enjoy the Den :-)
Guest Message by DevFuse

# A Game of Probabilities

18 replies to this topic

### #11 bonanova

bonanova

bonanova

• Moderator
• 6161 posts
• Gender:Male
• Location:New York

Posted 23 June 2012 - 01:55 AM

Spoiler for my guess

Generalizing to j and k,
Spoiler for looks like

• 0

Vidi vici veni.

### #12 bushindo

bushindo

Senior Member

• VIP
• 721 posts
• Gender:Male
• Location:Los Angeles, CA

Posted 24 June 2012 - 02:32 AM

You and I are playing a little game (that is in no way whatsoever related to any game played on this site *cough*). There are 12 objects, 4 of which have the property M. The goal is to choose an object with the property M.

However, you and I are playing by different rules. For each round, first there is the N phase, in which you secretly choose an object (write down its number or something), then there is the D phase, during which I pick an object and it is revealed whether or not the object has property M.

If the object I pick has the property M, I win, if not we keep going. During any N phase, after you choose an object, you can call 'STOP' and the game will end. Then you reveal your choices, and if at least one has the property M, you win. If none have the property M, I win.

Bonus: Generalize to a case of n objects, k of which have the property M.

Here's an approximative strategy,

Spoiler for

Edited by bushindo, 24 June 2012 - 02:41 AM.

• 0

### #13 Yoruichi-san

Yoruichi-san

"That Woman"

• Members
• 3395 posts
• Gender:Not Telling
• Location:Outside the box

Posted 24 June 2012 - 03:28 AM

Excellent, thank you Bushindo

Spoiler for In agreement

• 0
Women are definitely stronger. We are [Fe]males, after all...

Some of what makes me me is real, some of what makes me me is imaginary...I guess I'm just complex. ;P

<3 BBC's Sherlock, the series and the man. "Smart is the new sexy."

Chromatic Witch links now on my 'About Me' page! Episode 3 is finally here!

When life hands me lemons, I make invisible ink.

### #14 dpalmer

dpalmer

Newbie

• Members
• 7 posts

Posted 24 June 2012 - 04:37 AM

Spoiler for A solution and an extension

• 0

### #15 dpalmer

dpalmer

Newbie

• Members
• 7 posts

Posted 24 June 2012 - 04:41 AM

Oh, I was assuming that the choice was only secret for D, but N knows what he or she draws. Those are two different games.
• 0

### #16 voider

voider

Junior Member

• Members
• 50 posts
• Gender:Male

Posted 24 June 2012 - 05:01 AM

Consider the decision tree.
You can calculate the probability that you will win, when you predetermine on which round you will say STOP (if you survive until that round).
The first round probability is 4/12=0.33
Second is 0.397979...
This increases, until some round to say STOP where the probability of winning will begin to drop.
Obviously the "general solution" is to say STOP on the round that produces the overall highest probability of winning relative to the beginning of the decision tree.
My solution would be a computation, rather than a calculation. There are multiple ways of presenting it, but none of them would look nice.
• 0

### #17 voider

voider

Junior Member

• Members
• 50 posts
• Gender:Male

Posted 24 June 2012 - 05:08 AM

The best strategy for N is to yell STOP only and as soon as he or she recieves something with property M...

You can't say STOP after they've already won.
• 0

### #18 dpalmer

dpalmer

Newbie

• Members
• 7 posts

Posted 24 June 2012 - 05:17 AM

Spoiler for Another solution

• 0

### #19 dpalmer

dpalmer

Newbie

• Members
• 7 posts

Posted 24 June 2012 - 05:21 AM

You can't say STOP after they've already won.

Of course not, but it doesn't matter. If N knows what he or she draws, then the only reason to continue the game is if he or she draws not-M. Calling STOP at this point would ensure a loss. Continuing gives a chance of victory and a chance of loss. So clearly it's better to allow another round. This logic continues until k=n, at which point whoever draws next wins.
• 0

#### 0 user(s) are reading this topic

0 members, 0 guests, 0 anonymous users