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Ask The Wizard #345

When drawing 13 cards from a thoroughly shuffled deck, how many distinct card ranks should we anticipate seeing?

Suited89

The answer is 9.05037214885954 ranks.

This scenario is a classic example of a Markov Chain problem.

The table below outlines the anticipated quantity of card ranks, ranging from 0 to 4 cards, for any total number of cards drawn, from 1 through 52.

Expected Ranks by Cards Dealt

Cards 0 Ranks 1 Rank 2 Ranks 3 Ranks 4 Ranks Expected
Ranks
1 12.000000 1.000000 0.000000 0.000000 0.000000 1.000000
2 11.058824 1.882353 0.058824 0.000000 0.000000 1.941176
3 10.174118 2.654118 0.169412 0.002353 0.000000 2.825882
4 9.343577 3.322161 0.324994 0.009220 0.000048 3.656423
5 8.564946 3.893157 0.519088 0.022569 0.000240 4.435054
6 7.836014 4.373589 0.745498 0.044178 0.000720 5.163986
7 7.154622 4.769748 0.998319 0.075630 0.001681 5.845378
8 6.518655 5.087731 1.271933 0.118319 0.003361 6.481345
9 5.926050 5.333445 1.561008 0.173445 0.006050 7.073950
10 5.374790 5.512605 1.860504 0.242017 0.010084 7.625210
11 4.862905 5.630732 2.165666 0.324850 0.015846 8.137095
12 4.388475 5.693157 2.472029 0.422569 0.023770 8.611525
13 3.949628 5.705018 2.775414 0.535606 0.034334 9.050372
14 3.544538 5.671261 3.071933 0.664202 0.048067 9.455462
15 3.171429 5.596639 3.357983 0.808403 0.065546 9.828571
16 2.828571 5.485714 3.630252 0.968067 0.087395 10.171429
17 2.514286 5.342857 3.885714 1.142857 0.114286 10.485714
18 2.226939 5.172245 4.121633 1.332245 0.146939 10.773061
19 1.964946 4.977863 4.335558 1.535510 0.186122 11.035054
20 1.726771 4.763505 4.525330 1.751741 0.232653 11.273229
21 1.510924 4.532773 4.689076 1.979832 0.287395 11.489076
22 1.315966 4.289076 4.825210 2.218487 0.351261 11.684034
23 1.140504 4.035630 4.932437 2.466218 0.425210 11.859496
24 0.983193 3.775462 5.009748 2.721345 0.510252 12.016807
25 0.842737 3.511405 5.056423 2.981993 0.607443 12.157263
26 0.717887 3.246098 5.072029 3.246098 0.717887 12.282113
27 0.607443 2.981993 5.056423 3.511405 0.842737 12.392557
28 0.510252 2.721345 5.009748 3.775462 0.983193 12.489748
29 0.425210 2.466218 4.932437 4.035630 1.140504 12.574790
30 0.351261 2.218487 4.825210 4.289076 1.315966 12.648739
31 0.287395 1.979832 4.689076 4.532773 1.510924 12.712605
32 0.232653 1.751741 4.525330 4.763505 1.726771 12.767347
33 0.186122 1.535510 4.335558 4.977863 1.964946 12.813878
34 0.146939 1.332245 4.121633 5.172245 2.226939 12.853061
35 0.114286 1.142857 3.885714 5.342857 2.514286 12.885714
36 0.087395 0.968067 3.630252 5.485714 2.828571 12.912605
37 0.065546 0.808403 3.357983 5.596639 3.171429 12.934454
38 0.048067 0.664202 3.071933 5.671261 3.544538 12.951933
39 0.034334 0.535606 2.775414 5.705018 3.949628 12.965666
40 0.023770 0.422569 2.472029 5.693157 4.388475 12.976230
41 0.015846 0.324850 2.165666 5.630732 4.862905 12.984154
42 0.010084 0.242017 1.860504 5.512605 5.374790 12.989916
43 0.006050 0.173445 1.561008 5.333445 5.926050 12.993950
44 0.003361 0.118319 1.271933 5.087731 6.518655 12.996639
45 0.001681 0.075630 0.998319 4.769748 7.154622 12.998319
46 0.000720 0.044178 0.745498 4.373589 7.836014 12.999280
47 0.000240 0.022569 0.519088 3.893157 8.564946 12.999760
48 0.000048 0.009220 0.324994 3.322161 9.343577 12.999952
49 0.000000 0.002353 0.169412 2.654118 10.174118 13.000000
50 0.000000 0.000000 0.058824 1.882353 11.058824 13.000000
51 0.000000 0.000000 0.000000 1.000000 12.000000 13.000000
52 0.000000 0.000000 0.000000 0.000000 13.000000 13.000000

This inquiry is being posed and explored on my forum at Wizard of Vegas .

In your video poker programming tips , where you clarify that despite there being 2,598,960 potential starting hands in video poker using a 52-card deck, only 134,459 unique hand classifications are crucial for analysis.

I would like to know how many unique hand classifications exist when using two to six decks of cards?

anonymous

For this topic, I consulted my knowledgeable colleague, Gary Koehler, who specializes in video poker mathematics. Here are the insights he provided based on the number of decks:

Video Poker Classes of Hands

Decks Combinations Classes
1 2,598,960 134,459
2 91,962,520 202,735
3 721,656,936 208,143
4 3,091,033,296 208,468
5 9,525,431,552 208,481
6 23,856,384,552 208,481

If we roll five red dice and five blue dice, what is the likelihood that both sets yield the same outcome, irrespective of their order? For instance, achieving a result of 1-2-3-3-6 for both sets.

anonymous

3,557 / 559,872 = 0.006353238, or about 1 / 157.

The table below illustrates the various types of rolls possible:

  • The count of distinct combinations that can result in this roll. For a full house, for instance, there are six ways to select three dice showing the same number and five possibilities for the pair, which results in a total of 30 unique full house combinations.
  • The arrangements possible. For a full house, you can choose three of the five dice for the triplet in combin(5,3)=10 different ways, leaving the other two for the pair.
  • The total means of achieving a specific hand as a roll. This product results from the previous two columns. For example, there are 30 outcomes to form a full house multiplied by 10 arrangements, equating to 300 distinct ways to roll a full house.
  • The likelihood of attaining that hand. For the full house scenario, this translates to a probability of 300 out of 6.5= 0.038580.
  • The odds that both rolls yield the same result as well as match the specified hand. This is calculated by squaring the probability from the fourth column and dividing it by the second column. For example, the chance that two rolls are both a full house stands at 0.038580.2However, the chance that they match the same full house configuration is 1/30. Therefore, the probability of both rolls resulting in the same full house comes to 0.038580.2/30 = 0.00004961.

In the bottom right cell, the cumulative probability that both sets of rolls are identical is 0.00635324.

Matching Roll

Type
of Roll		 Different
Types		 Orders Total
Combinations		 Probability
One Roll		 Probability
Two Rolls
Five of a kind 6 1 6 0.00077160 0.00000010
Four of a kind 30 5 150 0.01929012 0.00001240
Full house 30 10 300 0.03858025 0.00004961
Three of a kind 60 20 1,200 0.15432099 0.00039692
Two pair 60 30 1,800 0.23148148 0.00089306
Pair 60 60 3,600 0.46296296 0.00357225
Five singletons 6 120 720 0.09259259 0.00142890
Total 7,776 1.00000000 0.00635324