Probabilities of drawing certain cards

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.

Comment on Probabilities of drawing certain cards

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Re: Probabilities of drawing certain cards by LanX (Saint) on Jan 06, 2023 at 18:41 UTC
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` [download] remaining cols left for the interested reader. :) ° Further reading: Binomial Coefficient Stochastics Combinatorics Cheers Rolf _{(addicted to the 𐍀𐌴𐍂𐌻 Programming Language :) Wikisyntax for the Monastery} update °) spoiler DB<74> sub A_and_B { bk($n-2,$h-2) / bk($n,$h) } DB<75> sub A_and_not_B { bk($n-2,$h-1) / bk($n,$h) } DB<76> sub A_xor_B { 2bk($n-2,$h-1) / bk($n,$h) } + # NB: 2 A_and_not_B DB<77> sub A_or_B { ( 2bk($n-2,$h-1) + bk($n-2,$h-2) ) / bk($n,$h) +} # NB: A_and_B + A_xor_B DB<78> printf ( "%2i ".("\t%2.1f"x4)."\n", $n=$_, map {100$_} A_and +_B, A_and_not_B, A_xor_B, A_or_B ) for 6..15 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 14 11.0 24.7 49.5 60.4 15 9.5 23.8 47.6 57.1 DB<79> [download]	[reply] [d/l] [select]
Re: Probabilities of drawing certain cards by LanX (Saint) on Jan 06, 2023 at 19:42 UTC
> 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> Cheers Rolf _{(addicted to the 𐍀𐌴𐍂𐌻 Programming Language :) Wikisyntax for the Monastery}	[reply] [d/l] [select]
Re: Probabilities of drawing certain cards by GrandFather (Saint) on Jan 06, 2023 at 21:55 UTC
And the Perl content is ...? Optimising for fewest key strokes only makes sense transmitting to Pluto or beyond	[reply]
Re^2: Probabilities of drawing certain cards by jdporter (Paladin) on Jan 06, 2023 at 22:41 UTC
This is a Meditation. It's just me meditating.	[reply]

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