Question

How can i find the the mean and variance of drawing cards? (details inside)

in a random deck of 52 regular cards, we begin to flip card by card.

note: regular deck, with 4 shapes(diamond,ace,..) and each shape has 13 cards(ace,2,3,.., queen, king) - regular cards.

how can i calculate the mean and variance of flipping cards until obtaining all the king and queen cards(8 cards totally, (king + queen) * 4 shapes(ace, diamonds,..)

thank you very much, really unsure about this one

Answer 1

`"The mean is "47.1111`
`"The variance is "23.0321`

Explanation:

`P("all K & Q in max n flips") = C(n,8)(8/52)(7/51)...(1/45)`
`= C(n,8) (8! 44!) / (52!)`
`= (C(n,8)) / (C(52,8))`

`"(we assume n >= 8 or otherwise C(n,k) = 0 if n < k)"`

`=> P("all K & Q in exactly n flips") = (C(n,8)-C(n-1,8))/(C(52,8))`

`=> E(n) = sum n*(C(n,8) - C(n-1,8))/(C(52,8))`
`= (1/(C(52,8))) sum (n*C(n,8) - n*C(n-1,8))`
`= (1/(C(52,8))) sum (n*C(n,8) - (n*(n-1)!)/(8! (n-9)!))`
`= (1/(C(52,8))) sum (n*C(n,8) - (n-8) (n!)/(8! (n-8)!))`
`= (1/(C(52,8))) sum (n*C(n,8) - (n-8)*C(n,8))`
`= (8/(C(52,8))) sum C(n,8)`
`= (1/(C(52,8)*7!)) sum n(n-1)(n-2)...(n-7)`
`= (1/(C(52,8)*7!)) Poly9(n)`

`"(it is a polynomial of degree 9 due to Faulhaber's formula)"`

`Poly9(n) = a_9 n^9 + a_8 n^8 + ... + a_1 n + a_0`

`Poly9(n) = 0, n < 8`
`=> a_0 = 0`
`=> a_9 + a_8 + ... + a_1 = 0`
`=> 512 a_9 + 256 a_8 + ... + 2 a_1 = 0`
`...`
`=> 7^9 a_9 + 7^8 a_8 + ... + 7 a_1 = 0`
`Poly9(8) = 8! = 40320`
`Poly9(9) = 9! + 8! = 403200`

`"The solution to this system of equations in 10 variables is"`

`a_9 = 1/9`
`a_8 = -3`
`a_7 = 98/3`
`a_6 = -182`
`a_5 = 1603/3`
`a_4 = -707`
`a_3 = -64/9`
`a_2 = 892`
`a_1 = -560`
`a_0 = 0`

`=> " mean = "(1/(C(52,8)*7!)) Poly9(52) = 47.111111`

`"The variance is even more calculation. I think it is too much"`
`"calculation for an exam. Still i give you the answer here :"`

`E(n^2) = (8/(C(52,8))) sum n * C(n,8)`
`= (1/(C(52,8)*7!)) sum n^2(n-1)(n-2)...(n-7)`
`= (1/(C(52,8)*7!)) Poly10(n)`

`Poly10(n) = a_10 n^10 + a_9 n^9 + a_8 n^8 + ... + a_1 n + a_0`

`Poly10(n) = 0, n < 8`

`Poly10(8) = 8*8! = 322560`
`Poly10(9) = 8*8! + 9*9! = 3588480`
`Poly10(10) = 8*8! + 9*9! +5*10! = 21732480`

`"The solution to the system of equations in 11 variables is here"`

`a_10 = 1/10`
`a_9 = -47/18`
`a_8 = 27`
`a_7 = -413/3`
`a_6 = 335.3`
`a_5 = -1253/6`
`a_4 = -572`
`a_3 = 7174/9`
`a_2 = 209.6`
`a_1 = -448`
`a_0 = 0`

`E(n^2) = (1/(C(52,8)*7!)) Poly10(52) = 2242.488888`
`=> " variance = "E(n^2) - (E(n))^2 = 23.0321`