A well-financed player, “Hal,” hired me some time ago to teach him 9/6 Triple Double Bonus Poker. This is not a particularly good game (98.15%), but it was the best he could find for the large stakes he wished to play.

I warned him, of course, that the game would likely be very expensive. He didn’t care that much. The excitement of hitting large royals and aces with a kicker (AWAK) would counterbalance any losing sessions he had. Okay. It’s not the way I want to play, but I was happy to teach him.

Fairly recently, he texted me the following picture:

Hal had been dealt four aces without a kicker on $25 Ten Play. This was a hand he paid $1,250 to play. This was the kind of hand he’d been waiting for. (In his dreams, of course, he was dealt the kicker with the aces for a cool million bucks.) He held the aces, hit the button, and ended up with two AWAK ($100,000 each) and eight of the hands didn’t improve ($20,000 apiece.) It added up to $360,000. Sweet!

Hal wanted to know if getting two AWAK was what he was supposed to get on this draw. I was not near my computer at the time, so I told him that it was a fairly common result and I’d get him more details once I got home.

To analyze this, I used the binomial distribution formula built into Excel. This is a good exercise to demonstrate how to use this formula.

The letters in blue across the top and the numbers in blue down the left side of the chart below have been added by me. On my computer, the row and column numbers show up outside the chart itself — so they don’t transfer when I cut and paste. I labeled them so you know what I’m talking about.

Rows B and G are identical. They indicate the number of successes (i.e., AWAK) out of ten tries, starting from AAAA.

Row C is the probability for each of the tries. There are 47 unseen cards, and 12 of them are kickers (namely four 2s, four 3s, and four 4s). Twelve chances out of 47 comes out to 0.255319, which is the number you see on every line in column C.

There will be formulas listed below. These formulas do not end with a period or a comma, even though when you put them into a sentence it appears necessary to add these punctuation marks.

Column D represents the chance for that total number of hits for each of the 11 possibilities — namely 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10. You can’t get less than zero. And with ten chances, the most hits you can get is ten. The formula for space D7 is =BINOMDIST(B7,10,C7,FALSE)

This formula must be entered exactly, or at least equivalently. (For example, you could have used 0.255319 instead of C7.) The B7 refers to the probability of number of hits. The 10 refers to ten chances. The C7 refers to the probability of each hit, and FALSE is the code word indicating exactly that number of hits.

Once you’ve entered in the formula in space D7, you can copy that all the way down the page. Excel will fill in the correct formula on each line.

Column E represents the probability of getting that number of AWAK, or fewer. (That is, for E10, it’s the probability of getting 3 AWAK plus the probability of getting 2 AWAK plus the probability of getting 2 AWAK plus the probability of getting 0 AWAK. For E7, fill in the formula =BINOMDIST(B7,10,C7,TRUE) and then you can copy downward. Notice that E8 is always the sum of D7 and D8. E9 is always the sum of D7, D8, and D9. Etc.

Column F is merely the inverse of column D. To create this, on space F7, enter the formula =1/D7

Even though the probability of hitting a lot of AWAK is listed as 0.000 in D15, D16, and D17 on the chart, the computer internally keeps this number to quite a few decimal places.

A |
B |
C |
D |
E |
F |
G |

2 |
||||||

3 |
||||||

4 |
Number of | Chance for | Chance for | Cumulative | Inverse | Number of |

5 |
Successes | 1 Hit | Total Hits | Chances | of Total Hits | Successes |

6 |
||||||

7 |
0 | 0.255319 | 0.052 | 0.052 | 19.07 | 0 |

8 |
1 | 0.255319 | 0.180 | 0.232 | 5.56 | 1 |

9 |
2 | 0.255319 | 0.277 | 0.510 | 3.60 | 2 |

10 |
3 | 0.255319 | 0.254 | 0.763 | 3.94 | 3 |

11 |
4 | 0.255319 | 0.152 | 0.916 | 6.57 | 4 |

12 |
5 | 0.255319 | 0.063 | 0.978 | 15.97 | 5 |

13 |
6 | 0.255319 | 0.018 | 0.996 | 55.90 | 6 |

14 |
7 | 0.255319 | 0.004 | 1.000 | 285.31 | 7 |

15 |
8 | 0.255319 | 0.000 | 1.000 | 2219.12 | 8 |

16 |
9 | 0.255319 | 0.000 | 1.000 | 29125.90 | 9 |

17 |
10 | 0.255319 | 0.000 | 1.000 | 849505.35 | 10 |

This chart is good for AWAK as well as 2s, 3s, or 4s with a kicker. It works just as well for Double Double Bonus as it does for Triple Double Bonus.

The chance of being dealt four aces without a kicker (rounded) is one in 72,200. There are three times as many chances of getting 2s, 3s, or 4s without a kicker, namely about one in 24,065.

When Hal looked at a version of this chart, he responded: “It looks like I’m supposed to get two or three hits when I try ten times.”

My response was that there is no “supposed to.” Ending up with two or three kickers are the most likely outcomes, but each of the 11 possibilities between 0 and 10 inclusive are possible.

If you want to label zero hits as “unlucky” and five hits as “very lucky,” go ahead. There is no precise definition of what luck is, but most people would agree with those labels in these cases