Say I have a deck of 52 cards. I draw a card and then return it to the deck, and the probability of drawing each card is 1/52 each time.

What is the expected value of the number of draws needed to draw every card in the deck at least once?

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- Dec 17th 2009, 09:37 PMcubrikalExpected value of draws to get all 52 cards in a deck?
Say I have a deck of 52 cards. I draw a card and then return it to the deck, and the probability of drawing each card is 1/52 each time.

What is the expected value of the number of draws needed to draw every card in the deck at least once? - Dec 18th 2009, 09:54 AMawkward
Cubrikal,

This is called the "coupon collector's problem" in reference to someone who wants to collect a complete set of coupons. In your case, you have 52 "coupons", i.e. cards.

You can find a discussion of the problem and a formula for the expected number of draws needed to get a complete set here:

Coupon collector's problem - Wikipedia, the free encyclopedia - Dec 18th 2009, 10:51 AMJoel
In probability theory, the

**coupon collector's problem**describes the "collect all coupons and win" contests. It asks the following question: Suppose that there are*n*coupons, from which coupons are being collected with replacement. What is the probability that more than*t*sample trials are needed to collect all*n*coupons? The mathematical analysis of the problem reveals that the expected number of trials needed grows as*O*(*n*log(*n*)). For example, when*n*= 50 it takes about 225 samples to collect all 50 coupons.

I am a bit confused by this

50log(50) =

**84.9485002**

how do i get 225? - Dec 18th 2009, 12:04 PMawkward
The "big Oh" formula is only meant to give an idea of the rate of growth. It is not meant for precise computation. Probably your best formula is

$\displaystyle E(T) \approx n \ln (n) + \gamma n + 1/2$

where $\displaystyle \gamma$ is the "Euler gamma constant", about 0.5772. - Dec 18th 2009, 04:01 PMcubrikal