Probabilities of drawing certain cards

[jdporter]

This shows the probabilities of drawing two specific cards (labeled A and B) in a hand of five cards, from decks of various sizes.

All numbers are percentages.

When the deck contains no 'draw' cards:

Deck size	A & B	A & !B	A ^ B	A | B
6	66.7	16.7	33.3	100.0
7	47.6	23.8	47.6	95.2
8	35.7	26.8	53.6	89.3
9	27.8	27.8	55.6	83.3
10	22.2	27.8	55.6	77.8
11	18.2	27.3	54.5	72.7
12	15.2	26.5	53.0	68.2
13	12.8	25.6	51.3	64.1

When the 'A' card causes Draw 1:

Deck size	A & B	A & !B	A ^ B	A | B
6	50.0	0.0	50.0	100.0
7	41.7	8.3	50.0	91.7
8	35.7	14.3	50.0	85.7
9	31.2	18.8	50.0	81.2
10	27.8	22.2	50.0	77.8
11	25.0	25.0	50.0	75.0
12	22.7	27.3	50.0	72.7

The probabilities of other combinations can be derived from those given in this table:

• A | !B (A or not B - possibly both conditions) is the inverse of A & !B.
• A ^ !B (A or not B but NOT both conditions) is the inverse of A ^ B.
• !A & !B (neither A nor B) is the inverse of A | B
• !A | !B (not A, or not B - possibly both conditions) is the inverse of A & B.
• !A ^ !B (not A, or not B, but NOT both conditions) is logically the same as A ^ B.

[LanX]

just for fun and cool uses for Perl, here the math for the first column

  DB<29> $h = 5

  DB<30> sub A_and_B { bk($n-2,$h-2) / bk($n,$h) }

  DB<31> sub bk { my ($n,$k) = @_; fac($n)/(fac($k)*fac($n-$k)) }

  DB<32> sub fac { my ($n) =@_; my $f=1; $f*=$_ for 2..$n; $f}

  DB<33> printf ( "%i %0.3f\n", $n=$_, A_and_B ) for 6..15
6 0.667
7 0.476
8 0.357
9 0.278
10 0.222
11 0.182
12 0.152
13 0.128
14 0.110
15 0.095

remaining cols left for the interested reader. :) °

Further reading:

• Binomial Coefficient
• Stochastics
• Combinatorics

update

°) spoiler

<Reveal this spoiler or all in this thread>

[LanX]

> The probabilities of other combinations can be derived from those given in this table:

This can be further condensed:

<Reveal this spoiler or all in this thread>

[GrandFather]

And the Perl content is ...?

[jdporter]

This is a Meditation. It's just me meditating.