Pepsi, Apple and Probability

[HarryKeogh]

I was never good at this so if someone could figure it out for me I'd appreciate it...

ok, Pepsi is running a promotion where if the bottle cap has a message written under it you win a free song download from Apple. The funny part is if you tilt the bottle just right you could read the winning (or non-winning) message. The odds of getting a winning cap are one in three.

the question: if I'm at the convenience store and tilt the first bottle and see that it's not a winner and I put it back and grab a second bottle (from among dozens of bottles) and pay for it (without checking the cap) what are the chances the second bottle is a winner? my guess is 1 in 2 but it's pretty much that...a guess. Can someone provide the correct answer along with the math behind it?

 

[Rolfe]

Is this like the two goats and the car all over again?

Because my brain swallowed itself trying to make sense of that one.

Rolfe.

 

[Jon_in_london]

the question: if I'm at the convenience store and tilt the first bottle and see that it's not a winner and I put it back and grab a second bottle (from among dozens of bottles) and pay for it (without checking the cap) what are the chances the second bottle is a winner? my guess is 1 in 2 but it's pretty much that...a guess. Can someone provide the correct answer along with the math behind it?

The odds of getting it will awlays be one in three.

 

[Dragon]

Re: Re: Pepsi, Apple and Probability

The odds of getting it will awlays be one in three.

No, they won't.
Think about it.
(Hint:- There is only a finite number of bottles.)

 

[patnray]

The odds would still be one in three since the the dozens of bottles on the shelf were all placed randomly and are independant of each other.

If you knew beforehand that there were 12 bottles on the self and you knew that there were 4 winners among them , then your odds would improve to 4 out of 11 after removing one non-winner. But you do not have this knowledge after checking just one bottle.

 

[wollery]

Let me expand on Jons' answer. If there are only three bottles and you can tell that one of them is not a winner then the probability that one of the other two is a winner is 1/2.

However there are millions of bottles. This means that knowing that one of them isn't a winner does not affect the odds of the next bottle you pick up. (It may be that none of the bottles on that shelf are winners.)

 

[patnray]

Re: Re: Re: Pepsi, Apple and Probability

No, they won't.
Think about it.
(Hint:- There is only a finite number of bottles.)

But you don't know how many winners are in the collection on the shelf. And the bottles are independant of each other.

If you checked every bottle on the shelf except one, and all the bottles you check were losers, what are the odds that the last bottle is a winner? Still 1 in 3.

 

[The Don]

[pedant mode] If there are exactly 1/3 of each and you have eliminated one, even if there are millions haven't the odds shifted, albeit in a miniscule fashion, towards 1/2[/pedant mode]

If there are millions of bottles then you have made no MATERIAL difference to your odds of winning.

 

[Valmorian]

Let me expand on Jons' answer. If there are only three bottles and you can tell that one of them is not a winner then the probability that one of the other two is a winner is 1/2.

Huh? So you're saying that if you can tell that two of them are not winners then the probability that the last one is a winner is 1/1, or 1?

I don't think so.

EDIT: Unless you mean there are only 3 bottles in the entire contest (which there is not).

 

[(S)]

I think the basis of the 1-in-2 guess is the idea that, after three tries, you're /guaranteed/ to win, because the odds are 1-in-3. As a matter of fact, you only have a 19/27 chance of winning after three tries. [A 2/3 chance of not winning on any particular draw. 2/3 * 2/3 * 2/3 = 8 / 27 chance of not winning at all after three. 1 - 8 / 27 = 19 / 27 chance of winning.] That's only 70%, for those of us who don't like fractions.

 

[HarryKeogh]

[pedant mode] If there are exactly 1/3 of each and you have eliminated one, even if there are millions haven't the odds shifted, albeit in a miniscule fashion, towards 1/2[/pedant mode]

If there are millions of bottles then you have made no MATERIAL difference to your odds of winning.

so if there are 900,000 bottles and 300,000 winners are you saying that my odds have changed from 300,000 in 900,000 to 300,000 in 899,999?

 

[(S)]

Yes.

 

[Ladewig]

No, they won't.
Think about it.
(Hint:- There is only a finite number of bottles.)

I give special recognition to the first post that ever made me laugh and curse simultaneously. It's very hard to breathe when one does that.

.. As a matter of fact, you only have a 19/27 chance of winning after three tries. [A 2/3 chance of not winning on any particular draw. 2/3 * 2/3 * 2/3 = 8 / 27 chance of not winning at all after three. 1 - 8 / 27 = 19 / 27 chance of winning.] That's only 70%, for those of us who don't like fractions.

The formula [( 1- (2/3)^n) where n is number of tries] applies only if one assumes that no one les has been checking the shelf for winners. If other people do peek, then the chance of winning drops.

 

[Dragon]

so if there are 900,000 bottles and 300,000 winners are you saying that my odds have changed from 300,000 in 900,000 to 300,000 in 899,999?

Yup.
Also, if each store gets a fair allocation of winning bottles then eliminating a "loser" in any particular store has more of an effect - say 30 in 90 to 30 in 89.
Of course, as Ladewing says, if previous customers have been peeking ...
I suppose the best strategy would be to go to a store which has just had a fresh delivery.

Oh, and we all have too much time on our hands.

 

[(S)]

The ... well, I wouldn't call it /best/, but the 'optimal' strategy is to peek yourself ;-)

 

[scribble]

I was never good at this so if someone could figure it out for me I'd appreciate it...

I can answer without using math. Your chances are zero, because every other asshat in the universe was smart enough to keep looking and all the winning bottles are long gone. Sorry.

 

[scribble]

Just for kicks, I'll share my personal winning strategy with you folks. I've discovered a marvelous mathematical proof behind this strategy but it is lengthy and I can't fit it here.

Step 1. Get kazaa-lite or some other decent non-spyware-infested P2P software.
Step 2. Download whichever songs you like.
Step 3. Enjoy a nice Diet Coke, and sit sound in the knowledge that while you may be raping musical artists, at least you're giving Jack Valenti nightmares.

 

[Yahweh]

so if there are 900,000 bottles and 300,000 winners are you saying that my odds have changed from 300,000 in 900,000 to 300,000 in 899,999?

I like those odds!

 

[sorgoth]

It also gives you a pretty damn good excuse to drink pop.

 

[Jim_MDP]

Step 3. Enjoy a nice Diet Coke, and sit sound in the knowledge that while you may be raping musical artists, at least you're giving Jack Valenti nightmares.

This gave me a good laugh till I realized…

When Jack has nightmares, hes comes up with idiotic ideas. Such as eliminating 'screener' copies for Academy members. This would mean I could no longer download them from BT.

Hey… now I'M gonna have nightmares.
Nice move, ya mook.