collecting cards

**elenaas** · November 12th 2009, 10:34 AM

You collect trading cards, which have different scenes from a movie. For one movie, there are 90 different cards in the set, and you have all of them except the final scene. To try to get this card, you buy 10 packs of 8 cards each. All cards in the pack are different, and each of the cards is equally likely to be in a given pack. Find the probability that you will get at least one copy of the final scene.
HINT: think backwards.

**Twig** · November 12th 2009, 03:03 PM

Hi!

There are $\binom{89}{8}$ different combinations of cards that does not contain the final scene card. The total number of eight card combinations is $\binom{90}{8}$ .

We calculate the probability that you won´t get the final scene card:

$P(\mbox{ "You don´t get the final scene card" } = \left(\frac{\binom{89}{8}}{\binom{90}{8}}\right)^{ 10}$

Hence,

$P(\mbox{ "You get at least one copy of the final scene card "} = 1 - \left(\frac{\binom{89}{8}}{\binom{90}{8}}\right)^{ 10} \approx 0.606$

**elenaas** · November 13th 2009, 03:40 AM

Thank you!

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