Poker odds problem - statto required!

[Venari]

Okay.

I think this is simple, but maybe it's deceptively simple, because I was always rubbish at statistics.

I am playing draw poker (with no wildcards) with three friends. We were all dealt five cards, I discarded/replaced three (all of which were 9 or below, no aces), now my hand is:

King, Queen, Jack, Ten and Nine of Spades.

Nobody else discarded any cards, they played their dealt hand.

What is the chance of just one other person at the table having a better hand than me? (i.e. a Royal Flush.)

I think the answer is roughly 1 in 48.3million, but I'd appreciate it if someone else would do the run through of how they would calculate it to see what I've done wrong (if anything.)

Venari.

 

[MisterWeary]

A straight flush huh...there's no way you can let that go, I'm losing the house on that one

 

[Famine]

Okay.

I think this is simple, but maybe it's deceptively simple, because I was always rubbish at statistics.

I am playing draw poker with three friends. We were all dealt five cards, I discarded/replaced three, now my hand is:

King, Queen, Jack, Ten and Nine of Spades.

Nobody else discarded any cards, they played their dealt hand.

What is the chance of just one other person at the table having a better hand than me? (i.e. a Royal Flush.)

I think the answer is roughly 1 in 48.3million, but I'd appreciate it if someone else would do the run through of how they would calculate it to see what I've done wrong (if anything.)

Venari.

First you need to work out what constitutes a better hand - there's only four, and they're the four Royal Flushes. One of those isn't possible, since you hold four of the five required cards for it. So the only better hands are:
A♣ K♣ Q♣ J♣ 10♣
A♦ K♦ Q♦ J♦ 10♦
A♥ K♥ Q♥ J♥ 10♥

So there's only 3 possibilities, out of all possible poker hands, for a better hand than yours. There's 2,598,960 possible poker hands, of which your hand precludes the possibility of:
A♠ K♠ Q♠ J♠ 10♠
Q♠ J♠ 10♠ 9♠ 8♠
J♠ 10♠ 9♠ 8♠ 7♠
10♠ 9♠ 8♠ 7♠ 6♠
9♠ 8♠ 7♠ 6♠ 5♠
48x K♠ K♥ K♦ K♣+1
48x Q♠ Q♥ Q♦ Q♣+1
48x J♠ J♥ J♦ J♣+1
48x 10♠ 10♥ 10♦ 10♣+1
48x 9♠ 9♥ 9♦ 9♣+1

And a large variety of combinations of full houses, flushes, straights, pairs, three-of-a-kinds and high card that I daren't even try to calculate. However I believe the number of remaining combinations once your hand has been extracted is 1,533,939 (binomial coefficient - any 5 card hand from a combination of the 47 remaining cards).

So the probability of someone beating your hand is:
(number of hands which can beat yours remaining in play)/(number of hands remaining in play)
= 3/1,533,939
= 1/511,313

If we take account of the three cards you discarded, the number of remaining hands in the deck is 1,086,008, so the probability drops to 1/362,002.


But I may be wrong.

 

[Venari]

Sounds a bit more sensible that what I had in mind.

I was multiplying up (which I think I know, deep down, is wrong)

Chance of someone else having an ace:

3 in 44

Chance of them having the KQJ10 to match it:
1 in 43, 1 in 42, 1 in 41 and 1 in 40

Chance of no-one else having an ace

37 in 39 and 36 in 38

and thus, all possibilities of better hands now excluded, the rest of the cards in the hand could be anything: 37 in 37, 36 in 36 etc.

Multiply up, = 1 in48.3million.

I know thins is wrong somehow, which is why I asked.

Any other takers to confirm Famine's thought process?

V.

 

[Famine]

Sounds a bit more sensible that what I had in mind.

I was multiplying up (which I think I know, deep down, is wrong)

Chance of someone else having an ace:

3 in 44

If it were a head-to-head game (and presuming it to be a "scoring" Ace - not the Ace of Spades which is rendered useless by your hand)...

Otherwise it's (3/44)+(3/44)+(3/44) for a four player game - 9/44 or 1 in 4.8.


It's a bit more complex though, to my mind. Because not everyone has the same chance of getting the cards they need. Let's assume that you're the last person to receive a card...

1st Card
Person 1: 15 in 52 chance to receive a scoring card (A to 10, Hearts, Diamonds or Spades). Meets chance, gets a scoring heart.
Person 2: 10 in 51 chance to receive a scoring card (A to 10, Diamonds or Spades). Meets chance, gets a scoring diamond.
Person 3: 5 in 50 chance to receive a scoring card (A to 10, Club). Meets chance, gets a scoring club.
Venari: Card.

2nd Card
Person 1: 4 in 48 chance to receive a scoring card (Remaining A to 10, Hearts). Meets chance, gets a scoring heart.
Person 2: 4 in 47 chance to receive a scoring card (Remaining A to 10, Diamonds). Meets chance, gets a scoring diamond.
Person 3: 4 in 46 chance to receive a scoring card (Remaining A to 10, Clubs). Meets chance, gets a scoring club.
Venari: Card.

3rd Card
Person 1: 3 in 44 chance to receive a scoring card (Remaining A to 10, Hearts). Meets chance, gets a scoring heart.
Person 2: 3 in 43 chance to receive a scoring card (Remaining A to 10, Diamonds). Meets chance, gets a scoring diamond.
Person 3: 3 in 42 chance to receive a scoring card (Remaining A to 10, Clubs). Meets chance, gets a scoring club.
Venari: Card.

4th Card
Person 1: 2 in 40 chance to receive a scoring card (Remaining A to 10, Hearts). Meets chance, gets a scoring heart.
Person 2: 2 in 39 chance to receive a scoring card (Remaining A to 10, Diamonds). Meets chance, gets a scoring diamond.
Person 3: 2 in 38 chance to receive a scoring card (Remaining A to 10, Clubs). Meets chance, gets a scoring club.
Venari: Card.

5th Card
Person 1: 1 in 36 chance to receive a scoring card (Remaining A to 10, Hearts). Meets chance, gets a scoring heart.
Person 2: 1 in 35 chance to receive a scoring card (Remaining A to 10, Diamonds). Meets chance, gets a scoring diamond.
Person 3: 1 in 34 chance to receive a scoring card (Remaining A to 10, Clubs). Meets chance, gets a scoring club.
Venari: Card.


So, the odds of the players getting all five required scoring cards without changing are:
Person 1: (15/52)*(4/48)*(3/44)*(2/40)*(1/36) = 1/439,296
Person 2: (10/51)*(4/47)*(3/43)*(2/39)*(1/35) = 1/586,216
Person 3: (5/50)*(4/48)*(3/42)*(2/38)*(1/34) = 1/1,040,060

But that's not necessarily the same as the odds of any player having a better hand than you.

 

[Venari]

This is why I quit statistics.

 

[wfooshee]

You didn't say, but we're assuming none of the three discards were needed for one of those royal flushes. Also, no wild cards? Wild cards add the possibiliy of 5 of a kind, which beats even a royal flush.

 

[Venari]

True, I will edit.

The assumption was the discards were not the needed cards, and no wildcards.

V.

 

[daan]

statto required!

I presume its now a case of statto found?

 

[Venari]

..

 

[Venari]

I presume its now a case of statto found?

I have no comment on the relevance of Famine's instant magnetic attraction to this comment.

I think he's a very intelligent and helpful (nay, munificent) person and and I thank him from the bottom of my heart.

V.

 

[BobK]

Correct me if I'm wrong, but wouldn't it be 3 X 5!/(44X43X42X41X40)?

It's been too long since I messed with probability stuff like this.

 

[MP]

Although it's not needed anymore, here is the Wiki page for 5 card poker statistics. I got lost after reading the first paragraph (I'm terrible at Math), but I'm sure someone will find this page interesting.

http://en.wikipedia.org/wiki/Poker_statistics

Edit.

I'm curious Venari, did someone actually have a royal flush when you had the straight flush? Or do you just want to feel more secure next time you stumble upon a straight flush and start betting the car, house etc.? . Or was the question just a hypothetical?

 

[Venari]

Cheers for the formula BobK - and the link m_p13.

In answer to your question, it is hypothetical, although I did win a poker hand with a royal flush in sixth form, many, many years ago (before the war.)

In fact, I haven't really played poker since then (probably a good thing.)

V.

 

[Venari]

And I think BobK is right for the possibility of having the cards, but I'm still not sure somehow. But, nevermind, math rules!

BobK gets a +rep for that.

I confirmed it on the site mullen_power13 linked to, so a +rep for him too!

Fab. Cheers!

Hey mods, call in the dogs and put out the fire, this thread is closed!

Venari.