Math help
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4/52 * 1/51 * 1/50 * 1/49 * 1/48 * 8/47 * 3/46 * 2/45 * 1/44 * 9!/4!
4/52 is because there are four suits of 10. 1/51 is because there is only one J with the same suit. Same for 1/50 (Q), 1/49 (K) and 1/48 (A). 8/47 is because there are 8 choices of a card that can be a 4 of a kind (2-9) since 10,J,Q,K,A are already taken. 3/46 is one of the three remaining cars of the same rank, same for 2/45 and 1/44. Finally 9! (9 factorial = 9*8*7*6*5*4*3*2*1) is because the cards can appear in any order, and divide by 4! because the four of a kind involves 4 cards that are equivalent (i.e. the suits are not relevant unlike royal flush).
The numerator starts at 52 and goes down by 1 for each card because there are that many left in the card from which to choose.
Do the arithmetic and you get 2.17446E-09, or 1 in 459,884,425.
https://youtu.be/sPTHNr8s76w
I think the pattern would beQuote: jrblackieI have an equation question. If nine cards are dealt from a regular deck of cards what is the math to get a royal flush and four of a kind. What would the odds be. How is this equation written in a formula?
ZZZZ 10x,J,Q,K,A (I turned it around to be different)
4oak and a Royal
the ZZZZ is from 8 ranks only.
2 thru 9 as the 4oak can't B a 10 to A
8Choose1rank * 4suitsChoose4(4 suits and choose all 4)
C(8,1)*C(4,4) (in Wolfram Alpha)
8 * 1 = 8
now to the 10x
there are 4(four) 10s to select from.
so (4 choose 1) * (1choose1)^4 (the 4 other Royal cards)
4 * 1 = 4
4 * 8 = 32
32 / 52choose 9
32 / 3 679 075 400 =
4 / 459 884 425
about 1 in 114 971 106.3
different from another post by a factor of 4
maybe a 3rd will try
OF course, if this happened playing at home after they were set up that way
then that is a different story
Sally
Quote: mustangsallyI think the pattern would be
ZZZZ 10x,J,Q,K,A (I turned it around to be different)
4oak and a Royal
the ZZZZ is from 8 ranks only.
2 thru 9 as the 4oak can't B a 10 to A
Why not? Is, say, 10, J, Q, K of spades, four Aces, and any other card not both a royal flush and four of a kind?
Otherwise, I get what you get.
Would the order in which you pick, either 4oak or royal first matter?
the order does not matter.Quote: jrblackieSo, if you pick a royal in spades first, you still have the other suits available for the 4oak. Right?
Would the order in which you pick, either 4oak or royal first matter?
do you want the royal as 5 cards and the 4oak as 4 cards? that is 9 cards total.
Don suggested that a royal with the 4oak being one rank of the royal would be 8 cards. that would be more calculations
btw,
why the question?
Sally
I appreciate your help. I have a new slot game and thought that a 9 card combo progressive might be interesting. It looks like it would be on the order of MegaBucks. It might be too much for the NGC to allow.
Quote: jrblackieI appreciate your help. I have a new slot game and thought that a 9 card combo progressive might be interesting. It looks like it would be on the order of MegaBucks. It might be too much for the NGC to allow.
If it's a slot machine, that's different; either you have to allow for the same card to appear on two or more reels, or you need to explain the layout of the cards on what I assume are nine reels. You cannot make the value on a reel dependent upon the values on one or more other reels - at least, not in Nevada.
If it's a straightforward nine-card deal, and you do allow something like a royal flush and four Aces (which would be only 8 cards) to count, then the total number of winning deals is:
All nine cards used: 4 (suits for the royal) x 8 (ranks for the 4OAK) = 32
Eight cards used: 4 (suits for the royal) x 5 (ranks for the 4OAK) x 44 (possible cards for the "ninth card") = 880
The probability would be 912 / 3,679,075,400, or about 1 / 4,034,074. This is much better than the reported chances of winning the Megabucks jackpot.
Keep in mind that Nevada will allow odds greater than 100,000,000 - 1; however, if they are that high, the odds must be displayed on the machine.
sounds like a fun idea.Quote: jrblackieAlso, the deal is 9 cards for a royal and 4OAK using all 9 cards. Can't combine any of the 4OAK with the royal. I think it's close to the 459M - 1. I still have to look at NGC for regulations, but I hope you're right. Thank you for the info.
the Royal and 4oak is about 1 in 115 million as calculated before.
good luck to you
Sally
Quote: jrblackieIsn't MegaBucks around 51M -1? Also, the deal is 9 cards for a royal and 4OAK using all 9 cards. Can't combine any of the 4OAK with the royal. I think it's close to the 459M - 1. I still have to look at NGC for regulations, but I hope you're right. Thank you for the info.
Nevada Gaming Regulations Technical Standards for Gaming Devices and On-Line Slot Systems, 2.070:
Jackpot Odds. If the odds of hitting any advertised jackpot that is offered by a gaming device exceeds 100 million to one, the odds of the advertised jackpot must be prominently displayed on the award glass or video display.
Got another question.
What is the equation for obtaining a 6 card flush in the first six cards dealt?
I guess the first card doesn't matter. So 1 in 52. After that the suit matters. So the next should be 1 in 12, 1 in 11 and so forth. Is that right? And how is the equation written?
Thanks.
Quote: jrblackieHi all
Got another question.
What is the equation for obtaining a 6 card flush in the first six cards dealt?
I guess the first card doesn't matter. So 1 in 52. After that the suit matters. So the next should be 1 in 12, 1 in 11 and so forth. Is that right? And how is the equation written?
Thanks.
Not 1 in 52 to start, you start with 52 in 52, or just 1. Here’s how I would write it, as there are multiple ways:
(52/52) * (12/51) * (11/50) * (10/49) * (9/48) * (8/47) = 0.00033715613 which is about 1 in 2965.985
Call it one in 3k and have a great rest of your day!
