statistics question

[griz]

If you have a regular deck of 52 cards, randomly face down, on average how many cards do you have to turn over before you get one specific card?  My first thought was 26 but it seems like it should be more than that.

[CNYCacher]

Your first thought seems correct to me.

[AZRedhawk44]

I'm doubting this is right, but I wrote this little bit of iterative SQL code to find out:

Code: [Select]


declare @cards float
declare @x float
declare @y float

set @cards = 52
set @x = 0
set @y = 1

while @cards <> 0
begin

set @x = @x + (1/@cards)  --likelihood of getting the right card this time
print 'On the ' + cast(@y as varchar(10)) + ' turn: ' + cast((100/@cards) as varchar(10)) + '% likelihood of getting it right.'
print '    Net likelihood of getting it right in ' + cast(@y as varchar(10)) + ' turns = ' + cast(100* @x as varchar(10)) + '%.'
set @cards = @cards - 1  --decriment the number of cards remaining
set @y = @y + 1 --increase the turn counter

end


The results:

Code: [Select]

On the 1 turn: 1.92308% likelihood of getting it right.
    Net likelihood of getting it right in 1 turns = 1.92308%.
On the 2 turn: 1.96078% likelihood of getting it right.
    Net likelihood of getting it right in 2 turns = 3.88386%.
On the 3 turn: 2% likelihood of getting it right.
    Net likelihood of getting it right in 3 turns = 5.88386%.
On the 4 turn: 2.04082% likelihood of getting it right.
    Net likelihood of getting it right in 4 turns = 7.92468%.
On the 5 turn: 2.08333% likelihood of getting it right.
    Net likelihood of getting it right in 5 turns = 10.008%.
On the 6 turn: 2.12766% likelihood of getting it right.
    Net likelihood of getting it right in 6 turns = 12.1357%.
On the 7 turn: 2.17391% likelihood of getting it right.
    Net likelihood of getting it right in 7 turns = 14.3096%.
On the 8 turn: 2.22222% likelihood of getting it right.
    Net likelihood of getting it right in 8 turns = 16.5318%.
On the 9 turn: 2.27273% likelihood of getting it right.
    Net likelihood of getting it right in 9 turns = 18.8045%.
On the 10 turn: 2.32558% likelihood of getting it right.
    Net likelihood of getting it right in 10 turns = 21.1301%.
On the 11 turn: 2.38095% likelihood of getting it right.
    Net likelihood of getting it right in 11 turns = 23.5111%.
On the 12 turn: 2.43902% likelihood of getting it right.
    Net likelihood of getting it right in 12 turns = 25.9501%.
On the 13 turn: 2.5% likelihood of getting it right.
    Net likelihood of getting it right in 13 turns = 28.4501%.
On the 14 turn: 2.5641% likelihood of getting it right.
    Net likelihood of getting it right in 14 turns = 31.0142%.
On the 15 turn: 2.63158% likelihood of getting it right.
    Net likelihood of getting it right in 15 turns = 33.6458%.
On the 16 turn: 2.7027% likelihood of getting it right.
    Net likelihood of getting it right in 16 turns = 36.3485%.
On the 17 turn: 2.77778% likelihood of getting it right.
    Net likelihood of getting it right in 17 turns = 39.1263%.
On the 18 turn: 2.85714% likelihood of getting it right.
    Net likelihood of getting it right in 18 turns = 41.9834%.
On the 19 turn: 2.94118% likelihood of getting it right.
    Net likelihood of getting it right in 19 turns = 44.9246%.
On the 20 turn: 3.0303% likelihood of getting it right.
    Net likelihood of getting it right in 20 turns = 47.9549%.
On the 21 turn: 3.125% likelihood of getting it right.
    Net likelihood of getting it right in 21 turns = 51.0799%.
On the 22 turn: 3.22581% likelihood of getting it right.
    Net likelihood of getting it right in 22 turns = 54.3057%.
On the 23 turn: 3.33333% likelihood of getting it right.
    Net likelihood of getting it right in 23 turns = 57.639%.
On the 24 turn: 3.44828% likelihood of getting it right.
    Net likelihood of getting it right in 24 turns = 61.0873%.
On the 25 turn: 3.57143% likelihood of getting it right.
    Net likelihood of getting it right in 25 turns = 64.6587%.
On the 26 turn: 3.7037% likelihood of getting it right.
    Net likelihood of getting it right in 26 turns = 68.3624%.
On the 27 turn: 3.84615% likelihood of getting it right.
    Net likelihood of getting it right in 27 turns = 72.2086%.
On the 28 turn: 4% likelihood of getting it right.
    Net likelihood of getting it right in 28 turns = 76.2086%.
On the 29 turn: 4.16667% likelihood of getting it right.
    Net likelihood of getting it right in 29 turns = 80.3752%.
On the 30 turn: 4.34783% likelihood of getting it right.
    Net likelihood of getting it right in 30 turns = 84.7231%.
On the 31 turn: 4.54545% likelihood of getting it right.
    Net likelihood of getting it right in 31 turns = 89.2685%.
On the 32 turn: 4.7619% likelihood of getting it right.
    Net likelihood of getting it right in 32 turns = 94.0304%.
On the 33 turn: 5% likelihood of getting it right.
    Net likelihood of getting it right in 33 turns = 99.0304%.
On the 34 turn: 5.26316% likelihood of getting it right.
    Net likelihood of getting it right in 34 turns = 104.294%.
On the 35 turn: 5.55556% likelihood of getting it right.
    Net likelihood of getting it right in 35 turns = 109.849%.
On the 36 turn: 5.88235% likelihood of getting it right.
    Net likelihood of getting it right in 36 turns = 115.731%.
On the 37 turn: 6.25% likelihood of getting it right.
    Net likelihood of getting it right in 37 turns = 121.981%.
On the 38 turn: 6.66667% likelihood of getting it right.
    Net likelihood of getting it right in 38 turns = 128.648%.
On the 39 turn: 7.14286% likelihood of getting it right.
    Net likelihood of getting it right in 39 turns = 135.791%.
On the 40 turn: 7.69231% likelihood of getting it right.
    Net likelihood of getting it right in 40 turns = 143.483%.
On the 41 turn: 8.33333% likelihood of getting it right.
    Net likelihood of getting it right in 41 turns = 151.817%.
On the 42 turn: 9.09091% likelihood of getting it right.
    Net likelihood of getting it right in 42 turns = 160.908%.
On the 43 turn: 10% likelihood of getting it right.
    Net likelihood of getting it right in 43 turns = 170.908%.
On the 44 turn: 11.1111% likelihood of getting it right.
    Net likelihood of getting it right in 44 turns = 182.019%.
On the 45 turn: 12.5% likelihood of getting it right.
    Net likelihood of getting it right in 45 turns = 194.519%.
On the 46 turn: 14.2857% likelihood of getting it right.
    Net likelihood of getting it right in 46 turns = 208.804%.
On the 47 turn: 16.6667% likelihood of getting it right.
    Net likelihood of getting it right in 47 turns = 225.471%.
On the 48 turn: 20% likelihood of getting it right.
    Net likelihood of getting it right in 48 turns = 245.471%.
On the 49 turn: 25% likelihood of getting it right.
    Net likelihood of getting it right in 49 turns = 270.471%.
On the 50 turn: 33.3333% likelihood of getting it right.
    Net likelihood of getting it right in 50 turns = 303.804%.
On the 51 turn: 50% likelihood of getting it right.
    Net likelihood of getting it right in 51 turns = 353.804%.
On the 52 turn: 100% likelihood of getting it right.
    Net likelihood of getting it right in 52 turns = 453.804%.

This shows probability of 100% likelihood of a correct guess in 33 or 34 guesses.

There's probably something wrong with my routine above, though.

[AZRedhawk44]

Yup.  Revised below:

Code: [Select]

declare @cards float
declare @x float
declare @y float

set @cards = 52
set @x = 0
set @y = 1

while @cards <> 0
begin

set @x = @x + (1/@cards)  --likelihood of getting the right card this time
print 'On the ' + cast(@y as varchar(10)) + ' turn: ' + cast((100 * 1/@cards) as varchar(10)) + '% likelihood of getting it right.'
print '    Net likelihood of getting it right in ' + cast(@y as varchar(10)) + ' turns = ' + cast(100* @y/52 as varchar(10)) + '%.'
set @cards = @cards - 1  --decriment the number of cards remaining
set @y = @y + 1 --increase the turn counter

end

Code: [Select]

On the 1 turn: 1.92308% likelihood of getting it right.
    Net likelihood of getting it right in 1 turns = 1.92308%.
On the 2 turn: 1.96078% likelihood of getting it right.
    Net likelihood of getting it right in 2 turns = 3.84615%.
On the 3 turn: 2% likelihood of getting it right.
    Net likelihood of getting it right in 3 turns = 5.76923%.
On the 4 turn: 2.04082% likelihood of getting it right.
    Net likelihood of getting it right in 4 turns = 7.69231%.
On the 5 turn: 2.08333% likelihood of getting it right.
    Net likelihood of getting it right in 5 turns = 9.61538%.
On the 6 turn: 2.12766% likelihood of getting it right.
    Net likelihood of getting it right in 6 turns = 11.5385%.
On the 7 turn: 2.17391% likelihood of getting it right.
    Net likelihood of getting it right in 7 turns = 13.4615%.
On the 8 turn: 2.22222% likelihood of getting it right.
    Net likelihood of getting it right in 8 turns = 15.3846%.
On the 9 turn: 2.27273% likelihood of getting it right.
    Net likelihood of getting it right in 9 turns = 17.3077%.
On the 10 turn: 2.32558% likelihood of getting it right.
    Net likelihood of getting it right in 10 turns = 19.2308%.
On the 11 turn: 2.38095% likelihood of getting it right.
    Net likelihood of getting it right in 11 turns = 21.1538%.
On the 12 turn: 2.43902% likelihood of getting it right.
    Net likelihood of getting it right in 12 turns = 23.0769%.
On the 13 turn: 2.5% likelihood of getting it right.
    Net likelihood of getting it right in 13 turns = 25%.
On the 14 turn: 2.5641% likelihood of getting it right.
    Net likelihood of getting it right in 14 turns = 26.9231%.
On the 15 turn: 2.63158% likelihood of getting it right.
    Net likelihood of getting it right in 15 turns = 28.8462%.
On the 16 turn: 2.7027% likelihood of getting it right.
    Net likelihood of getting it right in 16 turns = 30.7692%.
On the 17 turn: 2.77778% likelihood of getting it right.
    Net likelihood of getting it right in 17 turns = 32.6923%.
On the 18 turn: 2.85714% likelihood of getting it right.
    Net likelihood of getting it right in 18 turns = 34.6154%.
On the 19 turn: 2.94118% likelihood of getting it right.
    Net likelihood of getting it right in 19 turns = 36.5385%.
On the 20 turn: 3.0303% likelihood of getting it right.
    Net likelihood of getting it right in 20 turns = 38.4615%.
On the 21 turn: 3.125% likelihood of getting it right.
    Net likelihood of getting it right in 21 turns = 40.3846%.
On the 22 turn: 3.22581% likelihood of getting it right.
    Net likelihood of getting it right in 22 turns = 42.3077%.
On the 23 turn: 3.33333% likelihood of getting it right.
    Net likelihood of getting it right in 23 turns = 44.2308%.
On the 24 turn: 3.44828% likelihood of getting it right.
    Net likelihood of getting it right in 24 turns = 46.1538%.
On the 25 turn: 3.57143% likelihood of getting it right.
    Net likelihood of getting it right in 25 turns = 48.0769%.
On the 26 turn: 3.7037% likelihood of getting it right.
    Net likelihood of getting it right in 26 turns = 50%.
On the 27 turn: 3.84615% likelihood of getting it right.
    Net likelihood of getting it right in 27 turns = 51.9231%.
On the 28 turn: 4% likelihood of getting it right.
    Net likelihood of getting it right in 28 turns = 53.8462%.
On the 29 turn: 4.16667% likelihood of getting it right.
    Net likelihood of getting it right in 29 turns = 55.7692%.
On the 30 turn: 4.34783% likelihood of getting it right.
    Net likelihood of getting it right in 30 turns = 57.6923%.
On the 31 turn: 4.54545% likelihood of getting it right.
    Net likelihood of getting it right in 31 turns = 59.6154%.
On the 32 turn: 4.7619% likelihood of getting it right.
    Net likelihood of getting it right in 32 turns = 61.5385%.
On the 33 turn: 5% likelihood of getting it right.
    Net likelihood of getting it right in 33 turns = 63.4615%.
On the 34 turn: 5.26316% likelihood of getting it right.
    Net likelihood of getting it right in 34 turns = 65.3846%.
On the 35 turn: 5.55556% likelihood of getting it right.
    Net likelihood of getting it right in 35 turns = 67.3077%.
On the 36 turn: 5.88235% likelihood of getting it right.
    Net likelihood of getting it right in 36 turns = 69.2308%.
On the 37 turn: 6.25% likelihood of getting it right.
    Net likelihood of getting it right in 37 turns = 71.1538%.
On the 38 turn: 6.66667% likelihood of getting it right.
    Net likelihood of getting it right in 38 turns = 73.0769%.
On the 39 turn: 7.14286% likelihood of getting it right.
    Net likelihood of getting it right in 39 turns = 75%.
On the 40 turn: 7.69231% likelihood of getting it right.
    Net likelihood of getting it right in 40 turns = 76.9231%.
On the 41 turn: 8.33333% likelihood of getting it right.
    Net likelihood of getting it right in 41 turns = 78.8462%.
On the 42 turn: 9.09091% likelihood of getting it right.
    Net likelihood of getting it right in 42 turns = 80.7692%.
On the 43 turn: 10% likelihood of getting it right.
    Net likelihood of getting it right in 43 turns = 82.6923%.
On the 44 turn: 11.1111% likelihood of getting it right.
    Net likelihood of getting it right in 44 turns = 84.6154%.
On the 45 turn: 12.5% likelihood of getting it right.
    Net likelihood of getting it right in 45 turns = 86.5385%.
On the 46 turn: 14.2857% likelihood of getting it right.
    Net likelihood of getting it right in 46 turns = 88.4615%.
On the 47 turn: 16.6667% likelihood of getting it right.
    Net likelihood of getting it right in 47 turns = 90.3846%.
On the 48 turn: 20% likelihood of getting it right.
    Net likelihood of getting it right in 48 turns = 92.3077%.
On the 49 turn: 25% likelihood of getting it right.
    Net likelihood of getting it right in 49 turns = 94.2308%.
On the 50 turn: 33.3333% likelihood of getting it right.
    Net likelihood of getting it right in 50 turns = 96.1538%.
On the 51 turn: 50% likelihood of getting it right.
    Net likelihood of getting it right in 51 turns = 98.0769%.
On the 52 turn: 100% likelihood of getting it right.
    Net likelihood of getting it right in 52 turns = 100%.

[makattak]

Your first thought seems correct to me.

↑This.

There's probably something wrong with my routine above, though.

And, to annoy those who complain about the one word post.... ↑This. (Edit: It seems you rectified the error already.)