How to calculate odds on a multi-card draw.

I initially put this question to the administrator, and they answered back a few minutes ago recommending I post it in the forum... so here goes :

My friends and I have been struggling with calculating the odds in
various cardgame situations. I managed to find a person with a PHD
in Statistics, but his answers/solutions did not hold up to testing,
and he won't help anymore... so I'm asking you for help. We have
several examples, but I'll stick with one to start with, and maybe ask
the 2nd question on a follow-up email... assuming we get this 1st
question solved.

Imagine you have a deck of 52 cards, face down on the table. The
odds of randomly pulling a spade on a single draw from the deck is 25%
(13 spades, 52 cards, easy). What if I pull out four random cards
from the deck. What are the odds that there is at least one spade
among them ? Same problem, but if I pull seven random cards from the
deck of 52, what are the odds that there is at least one spade among
the seven ? I need to see the math... the method used to derive the
odds of finding a spade in those cards.

I hope to hear back from you with good clear examples of how to
calculate it !

OK, I'll take a stab. I like math.

The reason why there are several answers, all of which are defensible, is that there are several approaches.

Let's play a game. Maybe you know this one? There's a purpose here, to demonstrate a principle.

Let's say that you have three pairs of hole cards. One pair is aces. one pair is kings. One pair is ace + king.

You randomly pick one of the six cards and it's an Ace. What are the odds that the other card of that pair is an ace?

Odysseus101 said:

OK, I'll take a stab. I like math.

The reason why there are several answers, all of which are defensible, is that there are several approaches.

Let's play a game. Maybe you know this one? There's a purpose here, to demonstrate a principle.

Let's say that you have three pairs of hole cards. One pair is aces. one pair is kings. One pair is ace + king.

You randomly pick one of the six cards and it's an Ace. What are the odds that the other card of that pair is an ace?

Click to expand...

I hate math, but this made me smile..even I got it

dbchristy said:

I hate math, but this made me smile..even I got it

Click to expand...

So what's your answer Christy, and why?

I hate math too, unless its poker math, so...

50/50

Fan_of_EP said:

I initially put this question to the administrator, and they answered back a few minutes ago recommending I post it in the forum... so here goes :

My friends and I have been struggling with calculating the odds in
various cardgame situations. I managed to find a person with a PHD
in Statistics, but his answers/solutions did not hold up to testing,
and he won't help anymore... so I'm asking you for help. We have
several examples, but I'll stick with one to start with, and maybe ask
the 2nd question on a follow-up email... assuming we get this 1st
question solved.

Imagine you have a deck of 52 cards, face down on the table. The
odds of randomly pulling a spade on a single draw from the deck is 25%
(13 spades, 52 cards, easy). What if I pull out four random cards
from the deck. What are the odds that there is at least one spade
among them ? Same problem, but if I pull seven random cards from the
deck of 52, what are the odds that there is at least one spade among
the seven ? I need to see the math... the method used to derive the
odds of finding a spade in those cards.

I hope to hear back from you with good clear examples of how to
calculate it !

Click to expand...

Wow! You take 4:20 seriously, eh?

Don't let math drive you so crazy. You'll end up like Sheldon Cooper.

CardDead711 said:

I hate math too, unless its poker math, so...

50/50

Click to expand...

That's a good answer, and arguably it's correct. As we study math and introductory combinatorics and optimization, we learn about discrete events. Once you've flipped one card, you use that information to recalculate your odds of predicting the result of another flip. We know that there are 4 aces of 52 cards, so if you get one ace in the hole, then there are 3 aces left of 50 cards. So the odds of the first flop card being an ace are 3 in 50. If it's not an ace, the odds of the second flop card are slightly improved to 3 in 49, and if it's not an ace then 3 in 48 for the third flop card. This is poker math, how to quickly figure out the approximate odds of getting your pair of aces on the flop.

As I recall, it was Bertrand Russell who in the early 20th century refined the relationships between philosophy/mathematics/logic. He didn't invent it all, but Russell was remarkable both for being not insane and for being a gifted speaker, teacher, and writer. Russell used my example of 3 pairs of cards (he used coins I think) to demonstrate that often the events which we think are discrete, or mutually exclusive, are not so.

According to Russell, you're hunting for the pair of aces so if you flip an ace you've flipped one of the three. For two of the three aces on the board, the other card of its pair is an ace, and one of the aces has a king with it. Russell would say that the flipping of the cards isn't mutually exclusive in the information you've gleaned. Rather than simply eliminating the pair of Kings by flipping an ace and thus making it a 50/50 that you picked the pair of aces, you're at 66.6% of flipping a second ace once you've flipped the first.

Thinking about it in this way leads different approaches of solving a problem, to Heisenberg's uncertainty principle and Shroedinger's cat, essentially saying that the unflipped cards literally have no value until they are flipped and thus the probability function (which is a wave) collapses and a single result gels. In quantum physics, it's impossible to determine the precise location of an electron and trying to do so affects the path of it; all you can do is create probabilities of where the electron might be along its path depending on your knowledge of its properties.

So back to OP's question. You have 52 cards laid out and you want to pick a spade on your first pick, so one in four. Does selecting four of the 52 cards into a smaller pile and then flipping one of those four change the odds? Does eliminating cards one at a time from the 52 until there's only one left, does that change the odds of the last remaining card being a spade? Does it matter if you look at each of the 51 cards you eliminate as you do so? Most people would say "of course not!" and they are probably right. But most people would also have said that flipping an Ace leads to the odds of a second ace flip being 50%.

So it's no wonder that OP is having a hard time sorting this out, and even with the help of a PhD in statistics the answers aren't clear. (However, I suspect that the PhD is an exaggeration because someone that smart could answer the questions easily I'm sure). I hope that OP comes back so we can discuss it.

Fan_of_EP said:

Imagine you have a deck of 52 cards, face down on the table. The
odds of randomly pulling a spade on a single draw from the deck is 25%
(13 spades, 52 cards, easy). What if I pull out four random cards
from the deck. What are the odds that there is at least one spade
among them ? Same problem, but if I pull seven random cards from the
deck of 52, what are the odds that there is at least one spade among
the seven ? I need to see the math... the method used to derive the
odds of finding a spade in those cards.

I hope to hear back from you with good clear examples of how to
calculate it !

Click to expand...

So let's leave the theory behind and look at the question most directly. Firstly, simplify it. It doesn't matter whether a card is an ace or two or six, so we can discard those parameters. Then, we discard the concept of the other three suits. There are only spades (13) and non-spades (39). The non spades can all be white cards and spades black. That doesn't change the odds of anything in this set up.

So, the odds of drawing one black card are 13/52, or 25%. The odds of drawing two black cards are 13/52 * 12/51, or 5.88%. No problem there. Similarly, the odds of drawing three black cards in a row are 1.29%, four in a row is 0.2641%, five in a row is 0.0495%, six in a row is 0.008429%, and seven in a row is 0.0010994%. Intuitively we know (at least if we play a lot of cards) that drawing seven spades in a row is extremely unlikely, but the odds are even lower than we'd probably guess. Even getting 4 in a row is only a quarter of one per cent.

So in holdem if you have hole spades, what are the odds of drawing three spades on the flop? That depends on what cards are taken out of the deck by the other hole cards. Here's where our approach yields to different results, and ties into OP's question.

Do we assume equal distribution among all hole cards including yours? Do we assume equal distribution among all hole cards other than yours? Or do we ignore all the other hole cards as if they don't exist? Do we assume that we're taking 5 cards out of the remains of the deck and flip from that, and if we do does that make a difference? These questions are the assumptions upon which you build your formula, and those answers are sometimes counter-intuitive. Which is why different results come from different approaches; it all comes back to Bertrand Russell.

In OP's question there is an answer and it's not super complicated, but a fundamental understanding of the question and of your assumptions is the key to getting that answer.

Poker is a game of math, but it is also a game of people.

If you don't understand the math you will make bad decisions, but discussions like this make me think of the Married with Children episode where the blond bimbo Kelly tried to remember a ton of information and when she learned something new she had to forget something else.

Don't get distracted by the math too much. Learn it, understand it, but in the middle of a hand your focus should be on the hand, (the villain, position, betting line, ect...) not trying to get exact figures.

Most players just use the Rule of 4 and 2. You count your outs and multiply them by 4 on the flop (with 2 cards to come) and by 2 on the turn (with 1 card to come) which gives pretty close to your odds of making your hand. Make sure you are getting a good count of your outs (don't count outs twice) and you will have a rough idea of where you are at.

Always remember, it is a game of incomplete information. Unless heads up, you will never know what was mucked and you could think you have better odds to make a hand than you do.

Thanks Odysseus101, Let me try to refine the question. A deck of 52 cards, face down on the table. I select four of them, discarding the remaining 48. I scoop up the four cards in front of me and flip them over at the same time. How do I calculate the odds that there is at least one spade among them ? Not just ONE spade among the four... there could be two, or three, or all four (an unlikely event as you pointed out), but the odds of ANY spade showing up in the four card blind pull. I figure it's a simple math problem, but haven't found the answer yet.

karl coakley said:

...

Most players just use the Rule of 4 and 2. You count your outs and multiply them by 4 on the flop (with 2 cards to come) and by 2 on the turn (with 1 card to come) which gives pretty close to your odds of making your hand. Make sure you are getting a good count of your outs (don't count outs twice) and you will have a rough idea of where you are at.

...

Click to expand...

+1 It doesn't get much simpler than this, when it comes to what I would call Tier I math, meaning estimating the probability that you will improve your hand. I use the rule of 4 and 2 combined with the memorization of some of the most common scenarios, such as a flush or straight draws post flop and turn, the chance of flopping a set when dealt a pocket pair, and maybe a few other scenarios. The next piece of the math puzzle is understanding pot odds, which can be a little tricky to grasp at first, but isn't too difficult if you work at it a little. Pot odds try to answer the question what do you do with the estimated probabilities calculated using the rule 4 and 2, already described above. Then there is Tier II math, again, my lingo, which is understanding situational poker. An example of Tier II math would be late in a tournament understanding how scooping up blinds and antes with proper bet sizing can pay off, even if you are folding a percentage of these hands due to 3-bets. Tier III math, in my book, are the myriad of HUD statistics out there aimed at further exploiting both table tendencies and villain tendencies, as well as understanding your own habits and potential leaks.

Final Anecdote:
The math portion of poker shouldn't be underestimated. It can help build confidence in both your approach and in your abilities. It can also help you deal with bad beats and bad runs, if you let it. For example, think of the free throw shooter in a basketball game who is making 75% of his free throws. As a fan, should you be frustrated if this shooter is missing a free throw every other time he is approaching the foul line for a two shot try? Not if you understand the math. Making 3 out of every 4 shots, on average, means you should expect to miss a free throw every other time you approach the foul line (assuming two shots, and many trials, in this example). Most people don't think of it this way, though. We tend to focus too much on the favored outcome. Similarly, if you are a two-to-one favorite in a hand, this means you are going to lose one hand in three. Yet, some folks react to these situations on the table as if it is some form of highway robbery or something, when they lose. So really, when evaluating RNGs and so forth, we are talking confidence intervals. Bad beats and bad runs are part of these intervals. Having said all this, humans are flawed, and programmers are human, so I am an advocate for certification and a watchdog approach to online poker. It's just easier to say than it is to do, obviously.

Fan_of_EP said:

Thanks Odysseus101, Let me try to refine the question. A deck of 52 cards, face down on the table. I select four of them, discarding the remaining 48. I scoop up the four cards in front of me and flip them over at the same time. How do I calculate the odds that there is at least one spade among them ? Not just ONE spade among the four... there could be two, or three, or all four (an unlikely event as you pointed out), but the odds of ANY spade showing up in the four card blind pull. I figure it's a simple math problem, but haven't found the answer yet.

Click to expand...

Glad to see you back. Let's look at your problem. Examine your assumptions and parameters to see if they actually make a difference.
I don't think it matters if you pick four cards out of the 52 and then flip them all at once. The answer is the same as whether or not you pick four random cards, and flip them one at a time or altogether. Can we agree on that? If not, why do you think that picking four cards and putting them in your hand and then flipping (from) them makes a difference?

OK, let's look at that specific situation. Out of four cards, you'll get five potential results: zero spades, one spade, two spades, three spades, or four spades. The odds of each of those five results must add up to 100%.

The easiest approach is to eliminate the one result of zero spades, which we don't want. To get that we multiply together 39/52 * 38/51 * 37/50 * 36/49 to get 30.38175. That means on a four card draw, the odds of getting no spades is 30.38%. Which means that the odds of getting any other result (at least one spade) is 100-30.38 or 69.62%. Does that make sense?

Same approach on a seven card draw. The odds of getting no spades at all is 39/52 * 38/51 * 37/50 * 36/49 * 35/48 * 34/47 * 33/46 = .114967953 if I hit my calculator buttons correctly. So, on a seven card draw the odds of at least one spade are 88.5%

This formula might work;

(13/52) + (13/51) + (13/50) + (13/49) = 1.0343261 which is 103% chance one will be a spade.

But we all know for a certainty, that occasionally we will whiff in the draw, and get no spades.

Which shows that you have a problem with no absolute answer.

Odysseus101 said:

Same approach on a seven card draw. The odds of getting no spades at all is 39/52 * 38/51 * 37/50 * 36/49 * 35/48 * 34/47 * 33/46 = .114967953 if I hit my calculator buttons correctly. So, on a seven card draw the odds of at least one spade are 88.5%

Click to expand...

Why are you multiplying those fractions?

Fan_of_EP said:

Thanks Odysseus101, Let me try to refine the question. A deck of 52 cards, face down on the table. I select four of them, discarding the remaining 48. I scoop up the four cards in front of me and flip them over at the same time. How do I calculate the odds that there is at least one spade among them ? Not just ONE spade among the four... there could be two, or three, or all four (an unlikely event as you pointed out), but the odds of ANY spade showing up in the four card blind pull. I figure it's a simple math problem, but haven't found the answer yet.

Click to expand...

Other than assuming a randomized shuffle, I think you can't go beyond the simple odds that you have 1/13 th of the deck, and want one of 4 variables.

It's gonna be 25%. Over the long haul, it will work out to 25%. Perhaps a better way to wonder about this would be to get into the statistics of doing this, perhaps with some fixed sample size, and achieving the expected long term results.

And yes, the Heisenberg thing is valid, and adds the uncertainty. Also it is useless here, except that it shows that we can probably expect a random card if the deck is properly shuffled.

dj11 said:

Why are you multiplying those fractions?

Click to expand...

I'm multiplying the fractions because that's how it's done in these situations. Sometimes they are added, but usually multiplied. As you showed above, if you add fractions sometimes you get a probability of over 100%.

Let's say you're flipping a coin. It's 1/2 for heads. So if you are doing more than one flip, you multiply rather than add.

Odds of a head on one flip = 1/2 or 50%

Odds of two heads on two flips = 1/2 * 1/2 = 1/4 or 25%

Odds of three heads on three flips = 1/2 * 1/2 * 1/2 =1/8 or 12.5%

Or use a six sided dice. Odds of rolling a one are 1/6 or 16.667%. Roll two die, and the odds of getting two 1s are 1/6 * 1/6 or 1/36. This makes sense intuitively, because you know that there are 36 possible outcomes when rolling two die, and double ones is 1 of those 36 outcomes. Make sense?

Right?

dj11 said:

Other than assuming a randomized shuffle, I think you can't go beyond the simple odds that you have 1/13 th of the deck, and want one of 4 variables.

It's gonna be 25%. Over the long haul, it will work out to 25%. Perhaps a better way to wonder about this would be to get into the statistics of doing this, perhaps with some fixed sample size, and achieving the expected long term results.

And yes, the Heisenberg thing is valid, and adds the uncertainty. Also it is useless here, except that it shows that we can probably expect a random card if the deck is properly shuffled.

Click to expand...

On a single draw, you're right. It will be 25% when you are aiming for 13 of 52 cards. But the odds change if you hit the first spade. The first spade is 13/52, or 25%. Then you're aiming for one of 12 of 51 cards, which is 12/51, or 23.52%.

And you multiply them if you are looking for the odds of getting two spades in a row from start: 13/52 * 12/51 = 5.88%.

These are fairly simple calculations because there's really only one way to approach it. My point about Heisenberg and Schrodinger is that when the problem gets more complicated we have to look closely at the assumptions upon which we base the problem; the wrong assumptions will lead to a wrong answer, or a different but perhaps equally right answer for different circumstances/assumptions.

Odysseus101 said:

I'm multiplying the fractions because that's how it's done in these situations. Sometimes they are added, but usually multiplied. As you showed above, if you add fractions sometimes you get a probability of over 100%.

Let's say you're flipping a coin. It's 1/2 for heads. So if you are doing more than one flip, you multiply rather than add.

Odds of a head on one flip = 1/2 or 50%

Odds of two heads on two flips = 1/2 * 1/2 = 1/4 or 25%

Odds of three heads on three flips = 1/2 * 1/2 * 1/2 =1/8 or 12.5%

Or use a six sided dice. Odds of rolling a one are 1/6 or 16.667%. Roll two die, and the odds of getting two 1s are 1/6 * 1/6 or 1/36. This makes sense intuitively, because you know that there are 36 possible outcomes when rolling two die, and double ones is 1 of those 36 outcomes. Make sense?

Right?

Click to expand...

Difference here is that we are trying to figure out the odds of getting 1 specific result (1 or more spades).

If you multiply fractions less than 1, the results shrink. So in the card game we're playing (find a spade) according to your method here, the results of multiplying those fractions is something way less than 5% of the time we find a spade in 4 cards.

My addition goes;

I get a chance,
I get another (slightly better) chance,
I get another (even better) chance
I get a last (still better) chance.

We got 4 opportunities (chances) at 25% or better each, to find the spade.

I think the ultimate solution will involve properly framing the question.

Odysseus101 said:

I'm multiplying the fractions because that's how it's done in these situations. Sometimes they are added, but usually multiplied. As you showed above, if you add fractions sometimes you get a probability of over 100%.

Let's say you're flipping a coin. It's 1/2 for heads. So if you are doing more than one flip, you multiply rather than add.

Odds of a head on one flip = 1/2 or 50%

Odds of two heads on two flips = 1/2 * 1/2 = 1/4 or 25%

Odds of three heads on three flips = 1/2 * 1/2 * 1/2 =1/8 or 12.5%

Or use a six sided dice. Odds of rolling a one are 1/6 or 16.667%. Roll two die, and the odds of getting two 1s are 1/6 * 1/6 or 1/36. This makes sense intuitively, because you know that there are 36 possible outcomes when rolling two die, and double ones is 1 of those 36 outcomes. Make sense?

Right?

Click to expand...

Yes, but now the certainty of this thread meaning nothing is increased. We started with a particular problem, and that problem has changed to the point where we no longer have just that problem, we have several.

So, back the the original problem;

Fan_of_EP said:

Imagine you have a deck of 52 cards, face down on the table. The
odds of randomly pulling a spade on a single draw from the deck is 25%
(13 spades, 52 cards, easy). What if I pull out four random cards
from the deck. What are the odds that there is at least one spade
among them ? Same problem, but if I pull seven random cards from the
deck of 52, what are the odds that there is at least one spade among
the seven ? I need to see the math... the method used to derive the
odds of finding a spade in those cards.

I hope to hear back from you with good clear examples of how to
calculate it !

Click to expand...

Approximately 103%

In this case the reducing odds are additive, and the math is (13/52) + (13/51) + (13/50) + (13/49) = 1.0343261 which is 103% chance one will be a spade. That formula can be open ended, but at some point it might be more useful to consider the odds against a spade showing, which will not be certain until the 40th card is pulled.


If this alone were a betting game, the house would only pay 97%

dj11 said:

Yes, but now the certainty of this thread meaning nothing is increased. We started with a particular problem, and that problem has changed to the point where we no longer have just that problem, we have several.

Click to expand...

Not at all, dj. With respect, OP is asking what are the odds of getting at least one spade on a draw of four before you draw any. If you recalculate the odds with each draw, then indeed it's a different problem with each draw. We don't add those individual probabilities, we multiply them when looking entirely into the future.

That's what Russell's problem I mentioned earlier (where you have three pairs of cards and you're looking for the odds of hitting a second ace) demonstrates. Sometimes the information gleaned as the events proceed affects the ultimate probability, so the events in the chain aren't as mutually exclusive as we sometimes assume.

You wrote "Difference here is that we are trying to figure out the odds of getting 1 specific result (1 or more spades)." and that's incorrect. Getting more than one spade is the sum of several results: getting one spade, getting two spades, getting three spades, and getting four spades. We could calculate each of those four results and add them up, but it's easier to calculate the odds of getting no spades (30.38%) and recognize that the other four results (at least one spade) added up will account for the other 69.62%.

Now, that is an easy calculation. It doesn't tells us the odds of hitting one, two, three, or four spades. But we don't need to know that since OP wants the odds of at least one spade.

Lastly, how can the odds of something be over 100%? It can't. Your approach of adding it all up will yield a result of much higher than 103% if you take it to seven draws. And let's take it to 13 draws; obviously the probability is utterly tiny of drawing all 13 spades in 13 draws, but if you add it all up it's much higher than 100%. That should tell you that your understanding of the problem is flawed. Not to be condescending, or mansplaining, or whatever the term is these days. Don't worry, you're not alone by a long shot. And I won't calculate the odds of that long shot.

Fan_of_EP said:

Thanks Odysseus101, Let me try to refine the question. A deck of 52 cards, face down on the table. I select four of them, discarding the remaining 48. I scoop up the four cards in front of me and flip them over at the same time. How do I calculate the odds that there is at least one spade among them ? Not just ONE spade among the four... there could be two, or three, or all four (an unlikely event as you pointed out), but the odds of ANY spade showing up in the four card blind pull. I figure it's a simple math problem, but haven't found the answer yet.

Click to expand...

Isn't it just the same as none of the cards being a spade and subtract that from 100%, therefore meaning at least 1 is a spade

=1 - 48/52 * 47/51 * 46/50 * 45/49 ~ 28.13%

rhombus said:

Isn't it just the same as none of the cards being a spade and subtract that from 100%, therefore meaning at least 1 is a spade

=1 - 48/52 * 47/51 * 46/50 * 45/49 ~ 28.13%

Click to expand...

I don't follow. 48/52 on your first draw is the probability of drawing neither a spade nor a not-spade. Am I missing something?

Odysseus101 said:

I don't follow. 48/52 on your first draw is the probability of drawing neither a spade nor a not-spade. Am I missing something?

Click to expand...

oops in all the excitement I was thing of a card i.e. 4

so maybe

=1 - 39/52 * 38/51 * 37/50 * 36/49 ~ 69.62

similar to when you want to calculate a flush draw with 2 remaining cards. Calculate not hitting and subtract from 100%

=1 - 38/47 * 37/46 0.349676226