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More ways to get Braingle...

Russian Roulette

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 

Puzzle ID:#17681
Fun:*** (2.6)
Difficulty:** (2.07)
Category:Probability
Submitted By:JMCLEOD****
Corrected By:cnmne

 

 

 



You are in a game of Russian Roulette with a revolver that has 3 bullets placed in three consecutive chambers. The cylinder of the gun will be spun once at the beginning of the game. Then, the gun will be passed between two players until it fires. Would you prefer to go first or second?




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Comments

jimbo*au*
May 19, 2004

By winning you mean "not dying" so I presume you prefer to go second unless you have a large insurance policy and lots of creditors.
sparkplug*
May 20, 2004

I would not play this kind of game ! But since we are just talking about your teaser, it is a great one. Not hard (just imagine the possibilities as a drawing or a table) yet it is a good one.
robertbbr
May 23, 2004

I wish you had told me in the teaser itself how many chambers a revolver has....I liked it though!
fishmedAus*
May 27, 2004

Why not just shoot the other guy and you win?
pi202**
Dec 16, 2004

Hey robertbbr! I wished the same thing, but then thought about it a bit more and realised I didn't really need to know unless you want to work out the exact probability. The thing to remember is that with only one bullet, it would be evens. Then notice that after the first shot has been fired, and no bullet comes out, the only bullet you need to worry about is the first one, so it becomes similar to the one bullet problem. Hence it's just the first person to have a go that suffers a disadvantage as a result of the increased number of bullets.
brandiMcBride20Aus*
Mar 02, 2006

fishmed,,,LOL.
brainjuice**
Mar 27, 2006

i guess right but i dunno the reason.. haha
flagator12
Apr 17, 2007

I think your answer is too complicated. You already have a better chance of living by going second because no mater what happens, you will shoot the same amount of times or less than the other person. FUN RIDDLE
scienth
Sep 05, 2008

It's scary either way.

In the first round your opponent has a 50% chance of living/dying if he takes the first shot. If your opponent takes the first shot and lives, you have a 2/3 chance of dying!

But there is only a 1/2 * 2/3 = 2/6 or 1/3 chance of that exact scenario happening.
opqpop
Sep 24, 2010

Too easy.
Streuther
Nov 29, 2013

Hey!

I worked out a formula for the probabilities given an unspecified number of chambers...as previously mentioned, it is always best to go second. As there are 3 bullets there must be at least 3 chambers. Lets do the trivial ones first: P1, P2 = Player 1, Player 2

3 Chambers: P1 Loss (Trivial)

4 Chambers: 3/4 Chance P1 Loss (3/4 Chance of P1 dying in first shot. Otherwise, P2 definitely loses.)

5 Chambers, lets draw the chambers, the next chamber fired is one up from the previous one and I have written the result for P1 firing that chamber first. B means chamber has bullet, E means it is empty.

B P1 Loss
B P1 Loss
B P1 Loss
E P2 Loss (P1 shoots empty, P2 gets the bullet)
E P1 Loss (P1 shoots empty, P2 shoots empty, P1 gets bullet)

So, P1 has 4/5 chance of losing.

We can just extend the above, For example 8 chambers: (The author has already did 6 )
(Nc = Number of chambers)

B P1 Loss
B P1 Loss
B P1 Loss
E P2 Loss
E P1 Loss
E P2 Loss
E P1 Loss
E P2 Loss

(The reader should be able to check that...)

There is a pattern here which will quite clearly continue. P1 Loses if they pick one of the 1st 3 chambers first, then it alternates.

So the probability of P1 losing is 3 plus half the number of remaining chambers, if the number of remaining chambers is odd we round down.
The number of remaining chambers is Nc-3, so half the number of remaining chambers is (Nc-3)/2.
Rounding down can be expressed by the floor() function.

SO Probability of P1 Losing is (For Nc > 2):

(3 + Floor((Nc-3)/2)))/2

This will always be greater than 0.5*, so P1 is always more likely to lose (Note: as the number of chambers increases, the probability will approach 0.5)


*To prove this, simplify the equation the odd and even Nc and it should be clear.



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