Two Simple Applications of the Binomial Distribution

A month or so ago I answered the Question of the Day for the Las Vegas Advisor and I answered it in terms of the Binomial Distribution. There were some comments posted after that QOD indicating that people wanted to know more about it. This is a lightly edited version of a 2012 article that I published should be sufficient to respond to those questions.

I receive a lot of mail asking such questions as, “If I am dealt four cards to the royal flush (such as A♥ K♥ Q♥ T♥) and I am playing Fifty Play, how many royals will I usually end up getting?” or, “I played more than 200,000 hands of Jacks or Better and only received three royal flushes. How unlucky was this?”

These are questions that are easily answered by use of the Binomial Distribution. For the exact definition and formula for this distribution, consult a college level textbook in probability and statistics. That’s not our agenda today, because that is far more complicated than is appropriate here. What I want to do here is tell you HOW to get the answer to the question.

You will need a personal computer with Microsoft Excel on it. A considerable number of you have personal computers these days, and Excel is one of the programs that is preloaded on most of them. If you don’t have a computer now, you won’t be able to create your own chart, but you will be able to follow the interpretation presented. And even if you don’t have a computer now, save this article until you do. Anybody serious about winning at video poker needs a computer and who knows, maybe someday you’ll break down and get one?

So, open a new Excel workbook and do the following:

In space A1, type:  50

In space B1, type:  =1/47

In space C1, type:  0

In space D1, type:  =BINOMDIST(C1,A1,B1,FALSE)

Now hit ENTER.  If you have done it right, the number 0.341185 should have appeared in space D1.

In space E1, type:  =BINOMDIST(C1,A1,B1,TRUE)

Now hit ENTER and the number 0.341185 should have appeared again in E1.

Now click on A1 and drag the cursor to B1.  Then release the cursor.  This should outline both A1 and B1.  Hit CTRL and C at the same time.  This will copy the contents of A1 and B1 to your clipboard.

Now click your cursor on A1 and move the cursor down to A6 and over to B6 so that A2-6 and B2-6 are in a blue box.  Hit CTRL and V at the same time.  This will paste the contents of A1 into A2-6 and the contents of B1 into B2-6.

Now do the following:

In space C2, type:  1

In space C3, type:  2

In space C4, type:  3

In space C5, type:  4

In space C6, type:  5

Copy D1 into D2-6 and E1 into E2-6 using the technique described previously.  When you are done, your worksheet should look like that shown below.  (I have chosen to format columns B, D, and E with six digits to the right of the decimal point.  That many digits are not needed in this example, but they will be useful in the second example.)

So, what does this all mean? The 50 in each of the A spaces represents how many trials. Since we are playing Fifty Play and have 50 pops at the royal, 50 is the number of trials. If for some reason you were only playing 27 lines, you’d put 27 in each of the spaces. The 0.021277 in the B is the probability for success each time. You have one chance in 47 of connecting because you started with 52 cards in the deck, saw five of them, and were left with 47 from which to draw the one card that would give you the royal.  When you divide 1 by 47 on your calculator, 0.021277 is the number you get. The numbers 0 through 5 in column C represent the number of successes (i.e. 0 royals, or 1 royal or 2 royals, etc.) 

Column D represents the probability of getting EXACTLY the number of royal flushes shown in column 6. So (rounding slightly) you have a 34.1% chance of getting exactly 0 royals, 37.1% chance of getting exactly 1 royal, 19.8% chance of getting 2 royals, 6.9% chance of getting 3 royals, 1.8% chance of getting 4 royals and 0.4% chance of getting 5 royals. 

Column E represents the probability of getting EXACTLY the number of royals shown, OR LESS. So E1 is equal to D1. E1 is equal to D1 + D2 and E6 is equal to D1 + D2 + D3 + D4 + D5 + D6. You can easily extend the chart past 5 royals, but since there is a 99.93% chance of getting 5 royals or fewer, the numbers will get quite small.

Now let’s do some interpreting. Having 50 chances at a 1-in-47 opportunity sounds like we SHOULD get exactly one royal. But as you can see, getting 1 royal is only slightly more likely than getting 0 royals (37.1% to 34.1%), and you have considerable chances at hitting more than 1 royal also. Although we would LIKE to hit 10 or more royals from this position, it is extremely unlikely that you will ever do so. (Being dealt 50 royals is a 1 in 650,000 chance. You are far more likely to get 50 royals than to receive between 10 and 49.)

Now how about our second problem? You played 200,000 hands and only received three royals. How unlikely was is this?

I suggest you save the previous Excel spreadsheet under a name such as “Fifty Play Analysis” (so you don’t lose it) and then rewrite over it. In A1, you enter 200000 (because that’s the number of trials), in B1 you enter 0.000025 (because that is 1 in 40,000, which is approximately how often you hit a royal in Jacks or Better). I would extend column C down five more spaces (i.e. 6 though 10). Copy A1 and B1 down 10 spaces each, and copy D1 and E1 through all ten spaces below and you are done. Your chart should look like what you see here:

You have a 26.5% (about one out of four times) chance of getting three or fewer royals in 200,000 trials. Usually (three out of four times) you’ll get more. Yes, you were somewhat unluckier than average this time, but not much. It could have been worse!

One of the surprising things about looking at the second chart is that even though 200,000 hands is almost exactly five royal cycles, you are almost equally likely (17.547% each) to receive four royals as five. And even though five royals is the average, you are a big underdog to receive exactly that number, although slightly more likely to receive that exact number than any other.