Probability Question - trading cards

[gtc]

Hello,

I am trying to work out the probability that a pack of trading cards contains new cards or cards that I already have.

I am assuming that there are 100 cards in the set and that I already have 50 cards. Each pack contains 5 cards and all cards in a pack are different.

If I define the variable X as the number of cards that I do not already have, then what is the probabilty that X = 0, X = 1 and so on?

I thought about treating the pack as 5 successive draws

In that case the probabilty that the first card is new is just 50/100 but the proabilty that the second card is new depends on the outcome of the first card drawn.

If the first card was new then the probabilty that the second card is also new is 49/99 as there are now 49 cards still to be collected and the second card can't be the same as the first card. So the probabilty that the first two cards are new is 50/100 times 49/99 or 0.247.

If I already have the first card then the probability that I already have the second card is 49/99, so the probability that the first two cards are ones that I already have is also 0.247. The probabilty that only 1 of the first two cards is new can be worked out as one minus the other probabilities.

Scaling this up to three, four and five cards will be complicated. Is there a simpler way?

 

[Dunstan]

# of unique card combinations = 100 choose 5 = 100! / (95! 5!) = 75,287,520

# of combinations with no new cards = 50 choose 5 = 50! / (45! 5!) = 2,118,760

Pr(X = 0) = 2,118,760 / 75,287,520 = .0281

# of combinations with exactly one new card = (50 choose 4) * (50 choose 1) = 230,300 * 50 = 11,515,000

Pr (X = 1) = 11,515,000 / 75,287,520 = .1529

etc.

 

[Dunstan]

Finishing up, Pr (X = 2) = .3189

The calculation for X = 3 is the same as X = 2, and X = 4 is the same as X = 1, and X = 5 is the same as X = 0

 

[gtc]

Thank you.

 

[politas]

Hello,

I am trying to work out the probability that a pack of trading cards contains new cards or cards that I already have.

I am assuming that there are 100 cards in the set and that I already have 50 cards. Each pack contains 5 cards and all cards in a pack are different.

Your calculations will always be flawed due to the artificial scarcity built into trading cards by the manufacturers. Some cards are far more common than others. The only way to discover the actual distribution of cards is to get the information from them.

 

[Admiral]

Your calculations will always be flawed due to the artificial scarcity built into trading cards by the manufacturers. Some cards are far more common than others. The only way to discover the actual distribution of cards is to get the information from them.

What kind of cards are we talking about? Magic? Baseball? Pokemon?

 

[gtc]

Good point about the frequency. I was assuming each card had the same frequency.

I am actually collecting Paninni world cup stickers.

I haven't noticed any artificial scarcity, the percentages of foils and team photos is at least as great as the percentage of ordinary players in my collection.

That said, it is possible to manipulate the odds as the top-most sticker in each pack is visible.

It is also possible to order specific stickers directly, which renders artificial scarcity moot.

Each pack of 5 stickers costs AUD1.25; each sticker ordered seperately costs AUD 1. Therefore, when the expected number of new stickers in a pack falls below 1.25, it is time to order the remainder directly (if you are a completist)

 

[empeake]

Which ones do you need?

 

[Jaggy Bunnet]

Good point about the frequency. I was assuming each card had the same frequency.

I am actually collecting Paninni world cup stickers.

I haven't noticed any artificial scarcity, the percentages of foils and team photos is at least as great as the percentage of ordinary players in my collection.

That said, it is possible to manipulate the odds as the top-most sticker in each pack is visible.

It is also possible to order specific stickers directly, which renders artificial scarcity moot.

Each pack of 5 stickers costs AUD1.25; each sticker ordered seperately costs AUD 1. Therefore, when the expected number of new stickers in a pack falls below 1.25, it is time to order the remainder directly (if you are a completist)

Tried E-bay?

 

[politas]

Good point about the frequency. I was assuming each card had the same frequency.

I am actually collecting Paninni world cup stickers.

I haven't noticed any artificial scarcity, the percentages of foils and team photos is at least as great as the percentage of ordinary players in my collection.

That said, it is possible to manipulate the odds as the top-most sticker in each pack is visible.

It is also possible to order specific stickers directly, which renders artificial scarcity moot.

Each pack of 5 stickers costs AUD1.25; each sticker ordered seperately costs AUD 1. Therefore, when the expected number of new stickers in a pack falls below 1.25, it is time to order the remainder directly (if you are a completist)

Wow, it sounds like they may not be building a false scarcity into the set, if you can order any individual sticker. Certainly most of the trading card sets out there do have false scarcities (look at the different prices for different cards on ebay to see the effect).