webayarea
webayarea
Joined: Nov 30, 2018
  • Threads: 1
  • Posts: 6
November 30th, 2018 at 9:28:21 PM permalink
Looking for help in figuring the odds for a simple card game.

With 10 cards available to select from, A-10,

What are odds/payout of the following card combinations

A) 3 cards selected.
- odds of 3 in a row (aka 3 card Straight)?
- ie 2,3,4 or 7,8,9
- is it 8 out of 140? Or 15:1?

B) 4 cards selected
- odds of 3 out of 4 ina row (aka 3 card Straight)?
- ie 2,3,7,4 or 7,8,2,9
- 5:1 range? need help here

C) 5 cards selected
- odds of 3 out of 5 in a row (aka 3 card straight)?
- ie 2,3,7,4,8 or 7,8,2,9,3
- 2:1 range? Need help here

Thanks!!!
DogHand
DogHand
Joined: Sep 24, 2011
  • Threads: 1
  • Posts: 403
Thanks for this post from:
webayarea
December 2nd, 2018 at 11:57:14 AM permalink
Quote: webayarea

Looking for help in figuring the odds for a simple card game.

With 10 cards available to select from, A-10,

What are odds/payout of the following card combinations

A) 3 cards selected.
- odds of 3 in a row (aka 3 card Straight)?
- ie 2,3,4 or 7,8,9
- is it 8 out of 140? Or 15:1?

B) 4 cards selected
- odds of 3 out of 4 ina row (aka 3 card Straight)?
- ie 2,3,7,4 or 7,8,2,9
- 5:1 range? need help here

C) 5 cards selected
- odds of 3 out of 5 in a row (aka 3 card straight)?
- ie 2,3,7,4,8 or 7,8,2,9,3
- 2:1 range? Need help here

Thanks!!!



webayarea,

I'll do the first two, and leave the third to you.

(A) If we consider only combinations, 8 3-card straights are possible: A-low, 2-low, ..., 8-low. Each combination has 6 permutations: for example, the 3-low straight can be drawn in any of these orders:

345, 354, 435, 453, 534, and 543.

So, the total number of ways to draw the straight is 8*6, while the total permutations is 10*9*8. Thus, the probability of a straight for a 3-card draw is

P(3) = (8*6)/(10*9*8) = 1/15, or 15:1 against, as you said.

(B) Now for a 4-card draw, we still have only 8 3-card straight combinations. However, each of these can be accompanied by 7 "4th" cards, and the "4th" card can be drawn first, second, third, or fourth. So the numerator is now (8*6*7*4), while the denominator (total permutations) is (10*9*8*7), so for a 4-card draw the 3-card straight probability is

P(4) = (8*6*7*4)/(10*9*8*7) = 4/15, or 15:4 against.

Hope this helps!

Dog Hand
webayarea
webayarea
Joined: Nov 30, 2018
  • Threads: 1
  • Posts: 6
December 2nd, 2018 at 1:51:20 PM permalink
Dog! Thanks so much! Have a great day!
gordonm888
gordonm888
Joined: Feb 18, 2015
  • Threads: 43
  • Posts: 2730
December 2nd, 2018 at 4:39:14 PM permalink
When you say:

"C) 5 cards selected
- odds of 3 out of 5 in a row (aka 3 card straight)?"

Does that mean exactly 3 in a row? or would a 4 or 5 card straight qualify?
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
webayarea
webayarea
Joined: Nov 30, 2018
  • Threads: 1
  • Posts: 6
December 3rd, 2018 at 10:23:31 AM permalink
That would be leading to my next questions;

Odds of a 4/4 straight and 4/5 straight and then, biggest Longshot of 5/5

Therefore payouts on a single unit bet would be:

3/3 = 15:1
3/4 = 3.75:1
3/5= TBD (guessing 3/2? Maybe 5/4?)
LONGSHOTS
4/4= TBD
4/5 = TBD
5/5= TBD
ThatDonGuy
ThatDonGuy
Joined: Jun 22, 2011
  • Threads: 95
  • Posts: 4463
December 3rd, 2018 at 4:59:31 PM permalink
Quote: DogHand

(B) Now for a 4-card draw, we still have only 8 3-card straight combinations. However, each of these can be accompanied by 7 "4th" cards, and the "4th" card can be drawn first, second, third, or fourth. So the numerator is now (8*6*7*4), while the denominator (total permutations) is (10*9*8*7), so for a 4-card draw the 3-card straight probability is

P(4) = (8*6*7*4)/(10*9*8*7) = 4/15, or 15:4 against.


You forgot to take two things into account:
First, that the three cards in the 3-card straight can be in any order;
Second, that a 4-card straight contains two 3-card straights, each of which you are counting separately.

I prefer using combinations (where order does not matter) instead of permutations.
For the 4-card hand, there are (10)C(4) = 210 different hands.
There are 8 3-card straights; for each one, there are 7 different cards for the "fourth card", so that is 56 - but again, I am counting the four-card straights twice.
(For example, 2-3-4 and 5 as a fourth card is 2-3-4-5, but 3-4-5 and 2 as a fourth card, which is counted separately, is also 2-3-4-5.)
Subtract 7 (the number of 4-card straights) from the number of hands with 3-card straights; the result is 49.
P(4) = 49 / 210 = 7 / 30, or odds of 23-7 against.

Also note that, say, a probability of 4/15 is odds of (15-4) - 4, or 11-4, against, not 15-4 against.
Otherwise, the odds of a coin toss being heads would be 2-1 against.
7craps
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • Posts: 1977
December 3rd, 2018 at 5:47:17 PM permalink
Quote: webayarea

Looking for help in figuring the odds for a simple card game.

With 10 cards available to select from, A-10,

one can make a list of all the combinations
and see which ones are the events you are after
a helpful site for that
https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html

end up being the numerator being the ones that satisfy your event
the denominator would be either Combin(10,3) or Combin(10,4) or Combin(10,4) (120,210,252)

the math can get messy when one over-counts events
I made this list
see if it helps out
List has 120 entries.
1,2,3
1,2,4
1,2,5
1,2,6
1,2,7
1,2,8
1,2,9
1,2,10
1,3,4
1,3,5
1,3,6
1,3,7
1,3,8
1,3,9
1,3,10
1,4,5
1,4,6
1,4,7
1,4,8
1,4,9
1,4,10
1,5,6
1,5,7
1,5,8
1,5,9
1,5,10
1,6,7
1,6,8
1,6,9
1,6,10
1,7,8
1,7,9
1,7,10
1,8,9
1,8,10
1,9,10
2,3,4
2,3,5
2,3,6
2,3,7
2,3,8
2,3,9
2,3,10
2,4,5
2,4,6
2,4,7
2,4,8
2,4,9
2,4,10
2,5,6
2,5,7
2,5,8
2,5,9
2,5,10
2,6,7
2,6,8
2,6,9
2,6,10
2,7,8
2,7,9
2,7,10
2,8,9
2,8,10
2,9,10
3,4,5
3,4,6
3,4,7
3,4,8
3,4,9
3,4,10
3,5,6
3,5,7
3,5,8
3,5,9
3,5,10
3,6,7
3,6,8
3,6,9
3,6,10
3,7,8
3,7,9
3,7,10
3,8,9
3,8,10
3,9,10
4,5,6
4,5,7
4,5,8
4,5,9
4,5,10
4,6,7
4,6,8
4,6,9
4,6,10
4,7,8
4,7,9
4,7,10
4,8,9
4,8,10
4,9,10
5,6,7
5,6,8
5,6,9
5,6,10
5,7,8
5,7,9
5,7,10
5,8,9
5,8,10
5,9,10
6,7,8
6,7,9
6,7,10
6,8,9
6,8,10
6,9,10
7,8,9
7,8,10
7,9,10
8,9,10


List has 210 entries.
1,2,3,4
1,2,3,5
1,2,3,6
1,2,3,7
1,2,3,8
1,2,3,9
1,2,3,10
1,2,4,5
1,2,4,6
1,2,4,7
1,2,4,8
1,2,4,9
1,2,4,10
1,2,5,6
1,2,5,7
1,2,5,8
1,2,5,9
1,2,5,10
1,2,6,7
1,2,6,8
1,2,6,9
1,2,6,10
1,2,7,8
1,2,7,9
1,2,7,10
1,2,8,9
1,2,8,10
1,2,9,10
1,3,4,5
1,3,4,6
1,3,4,7
1,3,4,8
1,3,4,9
1,3,4,10
1,3,5,6
1,3,5,7
1,3,5,8
1,3,5,9
1,3,5,10
1,3,6,7
1,3,6,8
1,3,6,9
1,3,6,10
1,3,7,8
1,3,7,9
1,3,7,10
1,3,8,9
1,3,8,10
1,3,9,10
1,4,5,6
1,4,5,7
1,4,5,8
1,4,5,9
1,4,5,10
1,4,6,7
1,4,6,8
1,4,6,9
1,4,6,10
1,4,7,8
1,4,7,9
1,4,7,10
1,4,8,9
1,4,8,10
1,4,9,10
1,5,6,7
1,5,6,8
1,5,6,9
1,5,6,10
1,5,7,8
1,5,7,9
1,5,7,10
1,5,8,9
1,5,8,10
1,5,9,10
1,6,7,8
1,6,7,9
1,6,7,10
1,6,8,9
1,6,8,10
1,6,9,10
1,7,8,9
1,7,8,10
1,7,9,10
1,8,9,10
2,3,4,5
2,3,4,6
2,3,4,7
2,3,4,8
2,3,4,9
2,3,4,10
2,3,5,6
2,3,5,7
2,3,5,8
2,3,5,9
2,3,5,10
2,3,6,7
2,3,6,8
2,3,6,9
2,3,6,10
2,3,7,8
2,3,7,9
2,3,7,10
2,3,8,9
2,3,8,10
2,3,9,10
2,4,5,6
2,4,5,7
2,4,5,8
2,4,5,9
2,4,5,10
2,4,6,7
2,4,6,8
2,4,6,9
2,4,6,10
2,4,7,8
2,4,7,9
2,4,7,10
2,4,8,9
2,4,8,10
2,4,9,10
2,5,6,7
2,5,6,8
2,5,6,9
2,5,6,10
2,5,7,8
2,5,7,9
2,5,7,10
2,5,8,9
2,5,8,10
2,5,9,10
2,6,7,8
2,6,7,9
2,6,7,10
2,6,8,9
2,6,8,10
2,6,9,10
2,7,8,9
2,7,8,10
2,7,9,10
2,8,9,10
3,4,5,6
3,4,5,7
3,4,5,8
3,4,5,9
3,4,5,10
3,4,6,7
3,4,6,8
3,4,6,9
3,4,6,10
3,4,7,8
3,4,7,9
3,4,7,10
3,4,8,9
3,4,8,10
3,4,9,10
3,5,6,7
3,5,6,8
3,5,6,9
3,5,6,10
3,5,7,8
3,5,7,9
3,5,7,10
3,5,8,9
3,5,8,10
3,5,9,10
3,6,7,8
3,6,7,9
3,6,7,10
3,6,8,9
3,6,8,10
3,6,9,10
3,7,8,9
3,7,8,10
3,7,9,10
3,8,9,10
4,5,6,7
4,5,6,8
4,5,6,9
4,5,6,10
4,5,7,8
4,5,7,9
4,5,7,10
4,5,8,9
4,5,8,10
4,5,9,10
4,6,7,8
4,6,7,9
4,6,7,10
4,6,8,9
4,6,8,10
4,6,9,10
4,7,8,9
4,7,8,10
4,7,9,10
4,8,9,10
5,6,7,8
5,6,7,9
5,6,7,10
5,6,8,9
5,6,8,10
5,6,9,10
5,7,8,9
5,7,8,10
5,7,9,10
5,8,9,10
6,7,8,9
6,7,8,10
6,7,9,10
6,8,9,10
7,8,9,10


List has 252 entries.
1,2,3,4,5
1,2,3,4,6
1,2,3,4,7
1,2,3,4,8
1,2,3,4,9
1,2,3,4,10
1,2,3,5,6
1,2,3,5,7
1,2,3,5,8
1,2,3,5,9
1,2,3,5,10
1,2,3,6,7
1,2,3,6,8
1,2,3,6,9
1,2,3,6,10
1,2,3,7,8
1,2,3,7,9
1,2,3,7,10
1,2,3,8,9
1,2,3,8,10
1,2,3,9,10
1,2,4,5,6
1,2,4,5,7
1,2,4,5,8
1,2,4,5,9
1,2,4,5,10
1,2,4,6,7
1,2,4,6,8
1,2,4,6,9
1,2,4,6,10
1,2,4,7,8
1,2,4,7,9
1,2,4,7,10
1,2,4,8,9
1,2,4,8,10
1,2,4,9,10
1,2,5,6,7
1,2,5,6,8
1,2,5,6,9
1,2,5,6,10
1,2,5,7,8
1,2,5,7,9
1,2,5,7,10
1,2,5,8,9
1,2,5,8,10
1,2,5,9,10
1,2,6,7,8
1,2,6,7,9
1,2,6,7,10
1,2,6,8,9
1,2,6,8,10
1,2,6,9,10
1,2,7,8,9
1,2,7,8,10
1,2,7,9,10
1,2,8,9,10
1,3,4,5,6
1,3,4,5,7
1,3,4,5,8
1,3,4,5,9
1,3,4,5,10
1,3,4,6,7
1,3,4,6,8
1,3,4,6,9
1,3,4,6,10
1,3,4,7,8
1,3,4,7,9
1,3,4,7,10
1,3,4,8,9
1,3,4,8,10
1,3,4,9,10
1,3,5,6,7
1,3,5,6,8
1,3,5,6,9
1,3,5,6,10
1,3,5,7,8
1,3,5,7,9
1,3,5,7,10
1,3,5,8,9
1,3,5,8,10
1,3,5,9,10
1,3,6,7,8
1,3,6,7,9
1,3,6,7,10
1,3,6,8,9
1,3,6,8,10
1,3,6,9,10
1,3,7,8,9
1,3,7,8,10
1,3,7,9,10
1,3,8,9,10
1,4,5,6,7
1,4,5,6,8
1,4,5,6,9
1,4,5,6,10
1,4,5,7,8
1,4,5,7,9
1,4,5,7,10
1,4,5,8,9
1,4,5,8,10
1,4,5,9,10
1,4,6,7,8
1,4,6,7,9
1,4,6,7,10
1,4,6,8,9
1,4,6,8,10
1,4,6,9,10
1,4,7,8,9
1,4,7,8,10
1,4,7,9,10
1,4,8,9,10
1,5,6,7,8
1,5,6,7,9
1,5,6,7,10
1,5,6,8,9
1,5,6,8,10
1,5,6,9,10
1,5,7,8,9
1,5,7,8,10
1,5,7,9,10
1,5,8,9,10
1,6,7,8,9
1,6,7,8,10
1,6,7,9,10
1,6,8,9,10
1,7,8,9,10
2,3,4,5,6
2,3,4,5,7
2,3,4,5,8
2,3,4,5,9
2,3,4,5,10
2,3,4,6,7
2,3,4,6,8
2,3,4,6,9
2,3,4,6,10
2,3,4,7,8
2,3,4,7,9
2,3,4,7,10
2,3,4,8,9
2,3,4,8,10
2,3,4,9,10
2,3,5,6,7
2,3,5,6,8
2,3,5,6,9
2,3,5,6,10
2,3,5,7,8
2,3,5,7,9
2,3,5,7,10
2,3,5,8,9
2,3,5,8,10
2,3,5,9,10
2,3,6,7,8
2,3,6,7,9
2,3,6,7,10
2,3,6,8,9
2,3,6,8,10
2,3,6,9,10
2,3,7,8,9
2,3,7,8,10
2,3,7,9,10
2,3,8,9,10
2,4,5,6,7
2,4,5,6,8
2,4,5,6,9
2,4,5,6,10
2,4,5,7,8
2,4,5,7,9
2,4,5,7,10
2,4,5,8,9
2,4,5,8,10
2,4,5,9,10
2,4,6,7,8
2,4,6,7,9
2,4,6,7,10
2,4,6,8,9
2,4,6,8,10
2,4,6,9,10
2,4,7,8,9
2,4,7,8,10
2,4,7,9,10
2,4,8,9,10
2,5,6,7,8
2,5,6,7,9
2,5,6,7,10
2,5,6,8,9
2,5,6,8,10
2,5,6,9,10
2,5,7,8,9
2,5,7,8,10
2,5,7,9,10
2,5,8,9,10
2,6,7,8,9
2,6,7,8,10
2,6,7,9,10
2,6,8,9,10
2,7,8,9,10
3,4,5,6,7
3,4,5,6,8
3,4,5,6,9
3,4,5,6,10
3,4,5,7,8
3,4,5,7,9
3,4,5,7,10
3,4,5,8,9
3,4,5,8,10
3,4,5,9,10
3,4,6,7,8
3,4,6,7,9
3,4,6,7,10
3,4,6,8,9
3,4,6,8,10
3,4,6,9,10
3,4,7,8,9
3,4,7,8,10
3,4,7,9,10
3,4,8,9,10
3,5,6,7,8
3,5,6,7,9
3,5,6,7,10
3,5,6,8,9
3,5,6,8,10
3,5,6,9,10
3,5,7,8,9
3,5,7,8,10
3,5,7,9,10
3,5,8,9,10
3,6,7,8,9
3,6,7,8,10
3,6,7,9,10
3,6,8,9,10
3,7,8,9,10
4,5,6,7,8
4,5,6,7,9
4,5,6,7,10
4,5,6,8,9
4,5,6,8,10
4,5,6,9,10
4,5,7,8,9
4,5,7,8,10
4,5,7,9,10
4,5,8,9,10
4,6,7,8,9
4,6,7,8,10
4,6,7,9,10
4,6,8,9,10
4,7,8,9,10
5,6,7,8,9
5,6,7,8,10
5,6,7,9,10
5,6,8,9,10
5,7,8,9,10
6,7,8,9,10
winsome johnny (not Win some johnny)
7craps
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • Posts: 1977
December 3rd, 2018 at 6:02:41 PM permalink
Quote: webayarea

That would be leading to my next questions;

Odds of a 4/4 straight and 4/5 straight and then, biggest Longshot of 5/5

Therefore payouts on a single unit bet would be:

3/3 = 15:1

looks to me you may be mixing probability and odds. they are not the same.
https://www.mathsisfun.com/definitions/odds.html
https://www.mathsisfun.com/definitions/probability.html

for 3/3
8/120 = probability of success = 1 in 15
so 14 no against 1 yes
14 to 1 against
14:1 would be a fair payoff

your 15:1 payoff would have an edge
unless that is what you are after

why all the straights?
winsome johnny (not Win some johnny)
webayarea
webayarea
Joined: Nov 30, 2018
  • Threads: 1
  • Posts: 6
December 4th, 2018 at 11:17:01 AM permalink
Thanks All. Lots to digest and I appreciate the input. Please keep it coming!

For the 15:1 payout point, I agree and see the house edge being here and possibly setting at 15 FOR 1 to be player fair.

Im interested in the 'straights stats' as I've been thinking of a table game that involves straight combinations between 3-5 among a random 5 card draw from dealer. (Main thing is dealer only has A-10; no faces.)

I would have these lined up as likely single unit side bets and Longshot bets as a secondary part to a more primary focused game between dealer and player around the 5 cards in another capacity.
unJon
unJon
Joined: Jul 1, 2018
  • Threads: 13
  • Posts: 2088
December 4th, 2018 at 11:23:06 AM permalink
Quote: webayarea

Thanks All. Lots to digest and I appreciate the input. Please keep it coming!

For the 15:1 payout point, I agree and see the house edge being here and possibly setting at 15 FOR 1 to be player fair.

Im interested in the 'straights stats' as I've been thinking of a table game that involves straight combinations between 3-5 among a random 5 card draw from dealer. (Main thing is dealer only has A-10; no faces.)

I would have these lined up as likely single unit side bets and Longshot bets as a secondary part to a more primary focused game between dealer and player around the 5 cards in another capacity.

The calculations above assume the dealer only has 10 cards. If the dealer has multiple suits/decks on A-10 then the calculations above are not correct, and you need to specify the total number of A-10 decks.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.

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