+ Reply to Thread
Results 1 to 5 of 5

Thread: Challenge: Card drawing probability

  1. #1
    Points: 26, Level: 1
    Level completed: 52%, Points required for next Level: 24

    Posts
    3
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Question Challenge: Card drawing probability




    Hello everybody,

    This is my first post because this has been in my head and I couldn't hold it any longer! So here it goes:

    A friend of mine recently told me that with a deck of 52 cards, it is effectively impossible to get through the entire deck sequentially calling out "ace, two, three, ace, two, three etc.) without having one of the cards you drew matching the one you called.

    For example:
    You call: A 2 3 A 2 3 A 2 3
    You draw: 7 3 K 5 3 8 9 4 3 (bust!)

    You go bust, shuffle the cards and repeat.

    So, in light of knowing that there must be some winning combinations but obviously way smaller than the total amount of combinations (52!), I was wondering what is the probability of going through the deck of cards without going bust?

    I'm pretty sure it's related to conditional probability but I think using a software might be a better option.

    This has been in my mind for the last week as it piqued my interest as a simple yet effective probability problem. I thought about this with some friends and we couldn't come up with any approximation!

    Thank you!

  2. #2
    Points: 4,841, Level: 44
    Level completed: 46%, Points required for next Level: 109
    Con-Tester's Avatar
    Posts
    167
    Thanks
    3
    Thanked 59 Times in 57 Posts

    Re: Challenge: Card drawing probability

    A Monte Carlo simulation (110,000,000 trials) indicates the probabilities of going bust on drawing the first, second, third, etc. card. What’s perhaps a little counterintuitive here is that your chance of going bust is greatest on the very first card.

    The probability of not going bust at all is indeed very low at about 0.00818, or about 1 in 122.

    Code: 
    Card No.	P(Bust)
    1		0.076893664
    2		0.070856309
    3		0.065324627
    4		0.061534691
    5		0.056613664
    6		0.052106036
    7		0.048971273
    8		0.044997109
    9		0.041377409
    10		0.038879827
    11		0.035692882
    12		0.032733609
    13		0.030703064
    14		0.028123509
    15		0.025783191
    16		0.024155927
    17		0.022094164
    18		0.020206318
    19		0.018910109
    20		0.017293673
    21		0.015778855
    22		0.014740900
    23		0.013469982
    24		0.012283309
    25		0.011463027
    26		0.010425500
    27		0.009485545
    28		0.008849800
    29		0.008035573
    30		0.007307218
    31		0.006802091
    32		0.006170755
    33		0.005588500
    34		0.005224036
    35		0.004708727
    36		0.004263855
    37		0.003956445
    38		0.003580955
    39		0.003235282
    40		0.003005618
    41		0.002701982
    42		0.002444018
    43		0.002256636
    44		0.002036473
    45		0.001834173
    46		0.001695718
    47		0.001522209
    48		0.001369009
    49		0.001257355
    50		0.001127145
    51		0.001015691
    52		0.000934200
    
    Never		0.008178364
    One possible approach to an analytical solution would be to find the probability of going bust exactly on card n and summing these individual probabilities over 1 ≤ n ≤ 52. Subtracting this sum from 1 will give the probability of not going bust.

  3. #3
    Devorador de queso
    Points: 95,819, Level: 100
    Level completed: 0%, Points required for next Level: 0
    Awards:
    Posting AwardCommunity AwardDiscussion EnderFrequent Poster
    Dason's Avatar
    Location
    Tampa, FL
    Posts
    12,935
    Thanks
    307
    Thanked 2,629 Times in 2,245 Posts

    Re: Challenge: Card drawing probability

    Do you call your order before you see any of the cards or do you call after each card is shown with the option of choosing a different value based on what you just saw?
    I don't have emotions and sometimes that makes me very sad.

  4. #4
    Points: 26, Level: 1
    Level completed: 52%, Points required for next Level: 24

    Posts
    3
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Challenge: Card drawing probability

    Yes, that is completely true. I ran a simulation on Matlab and found out that the probability was around the same, 1 in every 123. Intuitive but still interesting to note is that the more cards you "call out", the higher the probabilities of not going bust. For example, calling out the whole deck (from Ace to King) increases your chance to almost 1 in every 62, almost double the chance from just doing the first three!

    Thank you for your help, I think we can consider this challenge completed!

  5. #5
    Points: 26, Level: 1
    Level completed: 52%, Points required for next Level: 24

    Posts
    3
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Challenge: Card drawing probability


    Quote Originally Posted by Dason View Post
    Do you call your order before you see any of the cards or do you call after each card is shown with the option of choosing a different value based on what you just saw?
    You call out before you see any of the cards, so in a way, when the deck is shuffled, you either have a winning combination or a losing one (0.82% of getting a winning one according to two simulations!).

+ Reply to Thread

           




Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts






Advertise on Talk Stats