Gambling is sometimes very fun. I like to win and I like to get money, and that sometimes happens while gambling. Other times I do not win and do not get money, which is less fun. So be it.

A while ago my friend Ryan won three games of Club Keno in a row. It’s important to me that Ryan knows how special he is, so I calculated the expected returns to the Michigan Lottery’s Club Keno game and made some graphs.

Rules of the game:

- For a single play, a player pays a dollar for a sheet of paper with a bunch of numbers on it.
- The players chooses anywhere from 1 to 10 unique numbers from 1 to 80. The number of numbers that the player picks is called the “spot.” For example, if the player chooses 11, 12, and 35, the player has chosen 3 numbers and is playing a 3 Spot game.
- The lottery randomly chooses 20 unique numbers from 1 to 80, and the player is paid based off of (i) how many numbers the player picked (the “spot”) and (ii) how many of the numbers that the player picked match those that the lottery picked (the “match”).
- Full prize details are available here.

Note:

- These calculations do not include the optional “The Jack” or “The Kicker” features.
- Pursuant to the “Common Lottery Myths” section on the “Responsible Gaming” page, I assume that draws are independent and identically distributed:

“Lottery drawings are completely random events and previous results have no bearing on future drawing outcomes. This means that for every lottery drawing, each number has an equal chance of being drawn and every set of numbers has the same chance of winning.”

(It’s worth noting here that the lottery website has numerous “hot number” features despite the fact that the draws are independent.)

(So be it.)

- Game rules and prize values are accurate as of July 15, 2018.
- Code available upon request.

The returns:

Game | Expected Gross Return | Expected Net Return |

1 Spot | 50.0¢ | -50.0¢ |

2 Spot | 66.1¢ | -33.9¢ |

3 Spot | 65.2¢ | -34.8¢ |

4 Spot | 64.9¢ | -35.1¢ |

5 Spot | 65.0¢ | -35.0¢ |

6 Spot | 64.8¢ | -35.2¢ |

7 Spot | 65.3¢ | -34.7¢ |

8 Spot | 64.7¢ | -35.3¢ |

9 Spot | 64.8¢ | -35.2¢ |

10 Spot | 63.7¢ | -36.3¢ |

So, you’re losing money on average in all cases, but your best return is playing 2 Spot and your worst return is playing 1 Spot. Never play 1 Spot.

For everything except the 1 Spot, you’re getting about a 35% expected loss. By way of comparison, Michigan slot machines are legally required to give no more than 25% expected loss (according to this source), and optimal Blackjack play in which the house uses one deck gives an expected loss of about 0.16% (according to this source). Of course, the house will use more than one deck, but the expected loss generally won’t go above 1%, meaning you can expect to lose over 35 times as much money playing Club Keno as you would if you perfectly played Blackjack.

I promised some graphs. Note that the axes differ from game to game. For all graphs I do not include the probability of winning nothing; it’s so much larger than the probability of winning something that the graphs would look boring (if you’re really clever you can just subtract the sum of the probabilities of winning from 100).

The 1 and 2 Spot games do not lead to interesting graphs, so I begin with the 3 spot game.

6 Spot and above offer very large prizes, but the probabilities of winning them are (literally) vanishingly small.

The 10 Spot game is interesting because they give a prize for matching exactly 0 numbers.

By the way, Ryan won three 7 Spot games in a row. I think those were the only three games he played, so the probability of that happening is about 1.3%.

Ryan, you’re very special.

Tony graduated in 2012 with majors in mathematics and economics. He now lives in Chicago and is pursuing graduate study in economics. He also has a very good cultural trivia podcast called “Here’s My Number, So Call Me Ishmael” available on Libsyn, iTunes, and Google Play.

Nice work. Found it interesting and useful.