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 February 18th, 2012, 04:30 AM #1 Newbie   Joined: Feb 2012 From: The Netherlands Posts: 2 Thanks: 0 Probability formula Hello, Probably a simple question here, but my math fu is weak: Say I have a standard deck of 52 playing cards (13 of each suit), and I randomly draw 7 cards. What are the odds I draw at least 3 Hearts cards? I'm mostly interested in how to calculate it, not so much the answer to this specific question. It's the 'at least' bit that stumps me. Any and all help much appreciated!
 February 18th, 2012, 06:58 AM #2 Senior Member   Joined: Oct 2011 From: Belgium Posts: 522 Thanks: 0 Re: Probability formula $\frac{\binom{13}{3}\binom{39}{4}+\binom{13}{4}\bin om{39}{3}+\binom{13}{5}\binom{39}{2}+\binom{13}{6} \binom{39}{1}+\binom{13}{7}\binom{39}{0}}{\binom{5 2}{7}}$ Clear when you see the formula?
February 18th, 2012, 07:14 AM   #3
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Re: Probability formula

Hello, Probably!

Quote:
 Say I have a standard deck of 52 playing cards (13 of each suit), and I randomly draw 7 cards. What is the probabiity that I draw at least 3 Hearts? It's the 'at least' bit that stumps me.

$\text{First, there are: }\:{52\choose7} \,=\,133,784,560\text{ possible 7-card hands.}$

"At least 3 Hearts" means: 3 Hearts or 4 Hearts or 5 Hearts or 6 Hearts or 7 Hearts.
[color=beige]. . [/color]We must compute each probability separately then add them.

To save time and energy, consider the opposite event: "at most 2 Hearts".
[color=beige]. . [/color]This means: 0 Hearts or 1 Heart or 2 Hearts.

0 Hearts
There are 13 Hearts and 39 Others.
We want 7 Others.
$\text{There are: }\:{39\choose7} \,=\,7,285,707\text{ ways.}$

1 Heart
We want 1 Heart, 6 Others.
$\text{There are: }\:{13\choose1}{39\choose6} \,=\,42,414,099\text{ ways.}$

2 Hearts
We want 2 Hearts, 5 Others.
$\text{There are: }\:{13\choose2}{39\choose5} \,=\,44,909,096\text{ ways.}$

$\text{Then there are: }\,7,285,707\,+\,42,414,099\,+\,44,909,046$
[color=beige]. . . . . . . . . . . . . . . [/color]$=\:96,608,852\text{ ways to get at most 2 Hearts}$

$\text{Hence, there are: }\,133,784,560\,-\,96,608,852 \:=\:37,175,708\text{ ways to get at least 3 Hearts}$

$\text{Therefore: }\:P(\text{at least 3 Hearts}) \;=\;\frac{37,175,708}{133,784,560} \;=\;0.277877417 \;\approx\;27.8\%$

February 18th, 2012, 08:47 AM   #4
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Joined: Feb 2012
From: The Netherlands

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Re: Probability formula

That's a great explanation, thanks!
I was trying to find a way to just plunk all the numbers into a single formula, so I could've been searching for a way to do that for ages with no result... It seems so obvious now, but I'd never have figured it out by myself.

Quote:
 Originally Posted by wnvl Clear when you see the formula?

February 18th, 2012, 07:57 PM   #5
Math Team

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Re: Probability formula

Quote:
 Originally Posted by soroban $\text{There are: }\:{39\choose7} \,=\,7,285,707\text{ ways.}$
I wrote simulation program and kept getting ~23.2% instead of Soroban's ~27.8%.
Couldn't figure out why.
Decided to check up on Soroban: 39choose7 = 15,380,937, not 7,285,707.

Make that change in Soroban's calculations and all is fine: .23231737... is the result, or ~23.2

Go stand in the corner, Soroban

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