01142015, 12:26 AM  #1 

Probability Question
In a standard 52card deck, what are the odds that out of five cards drawn at random, exactly three will be hearts? exactly four? all five? I think I've gotten the answer but I need to be absolutely 100% certain about it.

01142015, 12:37 AM  #2 
⠀
Join Date: Dec 2010
Location: Singapore, SG
Age: 22
Posts: 6,839

Re: Probability Question
There are 20 ways (5! / 3!) to have three hearts on the board, 5 ways (5! / 4!) to have 4 hearts on the board, 1 way to have 5 hearts on the board.
P(3 hearts) = (39/52 * 38/51 * 13/50 * 12/49 * 11/48) * 20 = 16.3085% P(4 hearts) = (39/52 * 13/51 * 12/50 * 11/49 * 10/48) * 5 = 1.07293% P(5 hearts) = (13/52 * 12/51 * 11/50 * 10/49 * 9/48) = 0.049519807% Edit: Actually not sure about the three hearts one... Last edited by EzExZeRo7497; 01142015 at 12:51 AM.. 
01142015, 12:54 AM  #3 
Wiki Staff
Join Date: Jun 2014
Posts: 307

Re: Probability Question
I'm getting 10 ways, (5 3) for 5 items in combinations of 3 = (5! / 3!2!)
P(exactly 3 hearts) = 0.08154 
01142015, 01:02 AM  #4 
Hunger Games Hunty

Re: Probability Question
10 ways is correct for 5 choose 3, yeah, so take eze's expression and multiply by 10 instead.
The other two look good. 
01142015, 01:03 AM  #5 
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Join Date: Dec 2010
Location: Singapore, SG
Age: 22
Posts: 6,839

Re: Probability Question
Yeah, my mistake. Should be 5C3, 5C4 and 5C5.

01142015, 01:43 AM  #6 

Re: Probability Question
Thanks everyone.

01142015, 05:23 AM  #7 
Kawaii Desu Ne?
Join Date: Dec 2007
Location: The Kawaiian Island~
Age: 26
Posts: 4,118

Re: Probability Question
Although this problem is already solved, here is an alternative solution which might be helpful for solving future similar problems.
So the probability of getting X hearts in a hand of 5 cards is equal to: The number of possible hands with X hearts  The number of possible hands overall The number of possible hands is simple, you have 52 cards and you have to pick 5, so there are C(52,5) such possible hands. Now consider the number of possible hands with X hearts. There are 13 hearts in the deck and 5213 nonhearts in the deck. Thus, you must pick X hearts from 13 possible hearts and 5X hearts from the 5213 nonhearts, i.e., there are C(13,X) ways to pick out the hearts for the hand and C(5213, 5X) ways of picking the nonhearts for the hand. So together, there are C(13,X) * C(5213, 5x) ways of forming a 5 card hand with X hearts. So, using the formula above, the probability of getting a hand with X hearts is: C(13,X) * C(5213, 5X)  C(52,5) 
01142015, 07:17 AM  #8  
Vice President Of TGB

Re: Probability Question
Fucking math nerds. And they said you wouldn't use math outside of school!
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Quote:


01142015, 10:01 AM  #9 
FFR Player

Re: Probability Question
you dont need math, you can just simulate 10,000 trials on excel like a pro
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01142015, 10:53 AM  #10 
FFR Veteran
Join Date: Aug 2013
Posts: 215

Re: Probability Question
Goddamn Statistics are killing me

01142015, 10:56 AM  #11  
Digital Dancing!
Join Date: Feb 2006
Location: 72 billion club, NE
Age: 27
Posts: 12,382

Re: Probability Question
Quote:
Brute force of 5,000,000 iterations gives the following: percentage of shuffles that resulted in 3 hearts in the first 5 cards of the deck: 8.1381% percentage of shuffles that resulted in 4 hearts in the first 5 cards of the deck: 1.0734% percentage of shuffles that resulted in 5 hearts in the first 5 cards of the deck: 0.0486%
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Last edited by rushyrulz; 01142015 at 11:32 AM.. 

01142015, 04:04 PM  #12 
FFR Player

Re: Probability Question
because i only have 10,000 decks of cards duh
I am also fairly surprised that five million trials is approaching the theoretical probabilities to only two significant digits. Statistics is weird!!!
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01162015, 04:06 PM  #13  
behanjc & me are <3'ers
Join Date: Jul 2006
Posts: 2,024

Re: Probability Question
Quote:
Let n be 5000000. Let p be the probability of getting exactly 3 hearts for one draw, or 2717/33320 (as shown already in this thread). Let z be the 1a/2 percentile of a standard normal distribution, and a be the error percentile. The confidence interval is p +/ z sqrt(p(1p)/n) For a 50% confidence level, z = 0.674. The confidence interval is between 8.146% and 8.163%. For a 99% confidence level, z = 2.576. The confidence interval is between 8.123% and 8.186%. In order for your confidence interval to shrink by a factor of 10 [due to sample size], or become closer to the theoretical value by one more significant digit, the second term needs to differ by a factor of 10. The variable n only appears once, and it is inside the root, so it's quite easy to see that the number of trials needs to be increased about 100fold. However, it should be noted that this is less accurate for small values of n and probabilities close to 0 and 1. Going backwards we can estimate 50,000 trials is accurate to a few percent, and 500 trials is accurate to a few tenpercent intervals. This is one of the rough approximations used for finding confidence intervals of binomial distributions. There are more accurate ones, but this one should give you a pretty good idea. Wrote some javascript code in like 2 minutes for anyone who wants to try: Code:
<script> // heartdraw.html var start = Date.now(); var deck = 52; var hearts = 13; var errors = []; // CHANGE THE FOLLOWING VARIABLES TO CHOOSE HOW MANY TRIALS AND DRAWS PER TRIAL var trials = 5; var totaldraws = 50000; // Test trials for (var n = 0; n < trials; n++) { var success = 0; for (var k = 0; k < totaldraws; k++) { if (Math.random() < 2717/33320) success++; // The below is commented out because it's brute force drawing and slower. // However, I kept it in in case anyone wants to try other draws at the cost of not needing to calculate the theoretical probabilties. /*var handhearts = 0; var decksize = 52; var heartsleft = 13; for (var i = 0; i < 5; i++) { if (Math.random() < heartsleft/decksize) { heartsleft; handhearts++; } decksize; } if (handhearts === 3) success++;*/ } // Display trial results if (trials <= 20) { // change the numerical value for display option var rate = success/totaldraws; var theoretical = 2717/33320; var error = (rate/theoretical1)*100; errors.push(error); console.log('Out of ' + totaldraws + ' draws, ' + success + ' were successes.'); if (trials <= 5) { // change the numerical value for display option console.log('Percentage of successful draws: ' + rate); console.log('Theoretical success rate: ' + theoretical); console.log('Percent error: ' + error); } } } // Check trial cumulative stats if (trials > 1) { var maxerror = errors[0]; var minerror = errors[0]; var sum = errors[0]; for (var n = 1; n < errors.length; n++) { if (errors[n] > maxerror) maxerror = errors[n]; else if (errors[n] < minerror) minerror = errors[n]; sum = sum + errors[n]; } var mean = sum/errors.length; console.log(' Trial overall results '); console.log('Mean trial error (percent): ' + mean); console.log('Range of trial errors (percent): ' + minerror + ' to ' + maxerror); } console.log('Time elapsed: ' + ((Date.now()start)/1000) + ' seconds.'); </script>
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Rhythm Simulation Guide Comments, criticism, suggestions, contributions, etc. are all welcome. Piano Etude Demon Fire sheet music Last edited by stargroup100; 01162015 at 04:12 PM.. 

01162015, 04:28 PM  #14 
Typical Bronzie

Re: Probability Question
I'd ballpark it in the range of like 0%33% for all of the above. Good enough, right?

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