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Ask The Wizard #345

If I were to deal 13 cards from a shuffled (presumed random) deck of cards, how many different ranks should I expect to see?

Suited89

The answer is 9.05037214885954 ranks.

 

This is a Markov Chain kind of problem if there ever was one.

The following table shows the expected number of ranks with 0 to 4 cards for all number of cards dealt from 1 to 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 question is asked and discussed in my forum at Wizard of Vegas.

In your video poker programming tips, you explain how that although there are 2,598,960 possible starting hands in video poker, with a 52-card deck, there are only 134,459 classes of hands necessary to analyze.

My question is how many classes of are there with two to six decks of cards?

anonymous

For this one, I turned to my esteemed colleague, Gary Koehler, who is an expert at video poker math. Here are his answers, according to 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

Five red dice and five blue dice are rolled. What is the probability the roll is the same for both dice, without regard to order. For example, both rolls are 1-2-3-3-6.

anonymous

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

 

The following the table shows for any type of roll:

  • The number of different ways this roll can be achieved. For example, for a full house, there are six combinations for the three of a kind and five left for the pair, for a total of 30 different full houses.
  • The number of orders. For example, for a full house, there are combin(5,3)=10 ways to choose three out of five dice for the three of a kind. The other two must have the pair.
  • The number of ways the given hand can be rolled. This is the product for the first two columns. For example, there are 30 * 10 = 300 ways to roll a full house.
  • The probability of the hand. For example, for a full house the probability is 300/65 = 0.038580.
  • The probability both rolls are the same and of the given hand. This is the probability from column four squared divided by the second column. For example, the probability two rolls are both a full house is 0.0385802. However, the probability they are the same house is 1/30. So, the probability both rolls are the same full house is 0.0385802/30 = 0.00004961.

The lower right cell shows the total probability both rolls are the same 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