Calculating probabilities
 

Ok I'm having a little bit of trouble with this one. See if you can help me out.

I bought a pack of baseball cards on ebay and received it today. I opened it up and got two autographs in the pack. The odds on one of the autographs was 1:33. The odds on the other was 1:597. The odds of getting the two would just be 33 X 597 = 19,701 right? Now I was thinking a little bit. On the back of the pack it stated that one card was inserted 1:1 (odds of one per pack). Now what would the be the chances of getting two of those in the same pack? You can't multiply the odds obviously.... I can't figure that out.

True the odds of both signatures being in the pack was 1 in 19,701
But I don't understand the second part of the question...

From my understanding of the question there is no mathmatical way to detrmine the odds of getting two autographs in one deck. Sounds like the maker of the cards makes sure every deck has at least one autographed card in a deck, but allows for some decks to get multiple autographed cards. That would only be determined by the rate at which the maker of the cards wants there to be multiple autographed cards in a deck. There is probably some law, or maybe a marketing strategy, that results in the makers publishing the odds for each autographed card to be in a deck. For any two autographed cards the odds would be different for getting two in a deck. The only thing you can say from the information given is that if you add up the odds for each card to be in a deck you would get one...

Actually after thinking about this a bit more, the odds either must be greater than 1:1 for getting an autographed card in a deck as you received two and therefore if the odds were exactly 1:1 there would be a deck out there without any... Though, maybe the maker doesn't guarantee one per deck...


Errr... I am getting way too confusing... I should have just said "I don't understand the second part of the question" like Silvy said... But that would have been boring, and I was in the mood to ramble on about something... Maybe I touched on the answer you were looking for. :)

For the second part:

Given
A = total number of all cards in the production run
c = total number of 1:1 cards in the deck
n = total number of cards in the pack

Odds of one out of all of the cards being the 1:1 card
P1 = 1 in A / (c*n)

Odds of one one out of the remaining cards being the 1:1 card
P2 = 1 in (A-1) / [(c-1)(n-1)]

Odds of two cards in the pack being the 1:1 card
P1 * P2

I'm not 100% sure of this answer, but it seems intuitive, and plugging test numbers into Excel seems to give appropriate answers.

Yikes, the explanation is more complicated than I could have imagined. I guess by the manufacturer saying the odds are 1:1, and you CAN get two in a pack, would imply that either the odds are correct and some packs would not have an autograph (I feel bad for the guy who paid $5 or more for that pack) or the odds are actually slightly higher than 1:1... maybe 1.00005:1, and they say it's 1:1 for simplicity. Thanks for the input guys!

It's actually a lot easier than that. One of the cards was inserted into your pack manually so that you were guaranteed an autograph. The other occurred randomly.

Since you don't know which card was inserted, either you got a random card at 1:33 odds or at 1:597. The former is more statistically likely, but either scenario is possible.