# Ask the Wizard #282

In the 2013 World Series of Poker final table, J.C. Tran was dealt 161 hands and said that not once did he receive a pocket pair and got ace-king only once. What is the probability of getting only one of these premium hands in 161 hands?

Ibeatyouraces

Probability of a pocket pair = 13*combin(4,2)/combin(52,2) = 5.88%.

Probability of AK = 4^{2}/combin(52,2)= 1.21%.

Probability of either = 5.88% + 1.21% = 7.09%.

Probability of NOT getting either = 100% -7.09% = 92.91%.

Probability of getting either once in 161 hands = 161*0.9291^{160}*0.0709^{1} = 1 in 11,268.

This question is discussed in my forum at Wizard of Vegas.

Assume that you do the following.

- Put a beaker on a scale.
- Glue a cork to the bottom of the inside of a beaker.
- Wait for glue to dry.
- Then fill it with water.

The glue is water-soluble. Eventually it will loosen and the cork will rise to the top. After the cork breaks free from the bottom of the beaker, but before it reaches the surface of the water, will the scale register more, less, or equal weight compared to when it was still glued to the bottom?

Anonymous

I claim that the answer is less. The way I would put it is the center of gravity is shifting downward as the cork moves upward, because water is denser than cork. The scale is measuring force applied against it. As the center of gravity moves down, force is being released, thus less is applied against the scale.

This question is discussed at my forum at Wizard of Vegas.

Thanks for your new section on parlay cards. I used your advice and got these lines (the market point spread is in parenthesis):

- Bills +3.5 (+3)
- Chargers +7.5 (+7)
- Cardinals -2.5 (-3)
- Dolphins -2.5 (-3)
- Bears +2.5 (0)

What is my advantage on this bet?

rudeboyoi

I'm going to assume the odds are 25 for 1, which is available on the half point card at the Golden Nugget, South Point, and William Hill sports book families.

The table below shows the line you got and the market price line.

First, the probability of an underdog beating the spread is 51.6%. That equates to a fair line of -106.6 on the underdog. So, you're getting 6.6 basis points on the underdogs, and losing them on the favorites.

Second, my table on buying a half point in the NFL shows the fair price to pay for each extra half point. For example, getting the extra half point off of 3 is worth laying -121.4, or 21.4 basis points.

The table breaks down how many basis points you're getting. For the Bears, I doubled the basis points for 1 and 2, since you're turning a loss into a win if you cross those numbers.

The table then converts the total basis points to a probability of winning. The formula is p = (100+b)/(200+b), where p = probability of winning and b = number of basis points.

The bottom row takes the product of each leg winning, for a probability of winning the parlay of 0.046751. At odds of 25 for 1, that bet has an expected return of 0.046751*25-1=0.168783. In other words, a 16.9% advantage. Well done!

### Rudeboyoi Parlay Card

Team | Parlay Card |
Fair | Underdog Basis Points |
Total Extra Points |
Total | Probability |
---|---|---|---|---|---|---|

Bills | 3.5 | 3 | 6.6 | 20.8 | 27.4 | 0.560246 |

Chargers | 7.5 | 7 | 6.6 | 11.9 | 18.5 | 0.542334 |

Cardinals | -2.5 | -3 | -6.6 | 20.8 | 14.2 | 0.533147 |

Dolphins | -2.5 | -3 | -6.6 | 20.8 | 14.2 | 0.533147 |

Bears | 2.5 | 0 | 0 | 18.0 | 18.0 | 0.541321 |

Product | 0.046751 |

William Hill is already accepting bets on the winner of the 2016 presidential election. What is the overall house edge on these bets?

Anonymous

The following table shows the choices, the odds they pay, the probability of winning, assuming it is a fair bet, and the adjusted probability to give each bet an equal house edge.

### 2106 Presidential Election Odds

Candidate | Pays | Fair Probability |
Adjusted Probability |
---|---|---|---|

Hillary Clinton | 2 | 0.333333 | 0.192293 |

Marco Rubio | 6 | 0.142857 | 0.082411 |

Jeb Bush | 9 | 0.100000 | 0.057688 |

Chris Christie | 10 | 0.090909 | 0.052444 |

Andrew Cuomo | 12 | 0.076923 | 0.044375 |

Paul Ryan | 16 | 0.058824 | 0.033934 |

Condoleeza Rice | 20 | 0.047619 | 0.027470 |

Deval Patrick | 20 | 0.047619 | 0.027470 |

Elizabeth Warren | 20 | 0.047619 | 0.027470 |

Mark Warner | 20 | 0.047619 | 0.027470 |

Martin O'Malley | 20 | 0.047619 | 0.027470 |

Rahm Emmanuel | 20 | 0.047619 | 0.027470 |

Rand Paul | 20 | 0.047619 | 0.027470 |

Bob McDonnell | 25 | 0.038462 | 0.022188 |

Cory Booker | 25 | 0.038462 | 0.022188 |

Rob Portman | 25 | 0.038462 | 0.022188 |

Jon Huntsman | 28 | 0.034483 | 0.019892 |

Joe Biden | 33 | 0.029412 | 0.016967 |

Michael Bloomberg | 33 | 0.029412 | 0.016967 |

Mitt Romney | 33 | 0.029412 | 0.016967 |

Sam Graves | 33 | 0.029412 | 0.016967 |

Susana Martinez | 33 | 0.029412 | 0.016967 |

Amy Klobuchar | 40 | 0.024390 | 0.014070 |

Scott Walker | 40 | 0.024390 | 0.014070 |

Bobby Jindal | 50 | 0.019608 | 0.011311 |

David Petraeus | 50 | 0.019608 | 0.011311 |

Mike Huckabee | 50 | 0.019608 | 0.011311 |

Rick Santorum | 50 | 0.019608 | 0.011311 |

Sarah Palin | 50 | 0.019608 | 0.011311 |

Mike Pence | 66 | 0.014925 | 0.008610 |

Dennis Kucinich | 100 | 0.009901 | 0.005712 |

Eric Cantor | 100 | 0.009901 | 0.005712 |

Evan Bayh | 100 | 0.009901 | 0.005712 |

Herman Cain | 100 | 0.009901 | 0.005712 |

John Kasich | 100 | 0.009901 | 0.005712 |

John Thune | 100 | 0.009901 | 0.005712 |

Julian Castro | 100 | 0.009901 | 0.005712 |

Kathleen Sebelius | 100 | 0.009901 | 0.005712 |

Kay Hagan | 100 | 0.009901 | 0.005712 |

Mia Love | 100 | 0.009901 | 0.005712 |

Michelle Obama | 100 | 0.009901 | 0.005712 |

Newt Gingrich | 100 | 0.009901 | 0.005712 |

Rick Perry | 100 | 0.009901 | 0.005712 |

Tim Kaine | 100 | 0.009901 | 0.005712 |

Total | 1.733465 | 1.000000 |

The overall expected return is the inverse of the sum of the fair probabilities. You can see the sum is 1.733465, so the overall expected return is 1/1.733465 = 57.69%. That would make the house edge 100% - 56.69% = 42.31%.

In pai gow poker, what would be the player advantage if he played only when his first card was an ace or a joker?

Eliot Jacobson

The following table shows the possible outcomes, assuming the player is not banking, and the Trump Plaza house way. The lower right cell shows a player advantage of 16.09%.

### First Card is an Ace or Joker

Outcome | Pays | Probability | Return |
---|---|---|---|

Win | 0.95 | 0.383010 | 0.363860 |

Tie | 0 | 0.413936 | 0.000000 |

Lose | -1 | 0.203054 | -0.203054 |

Total | 1.000000 | 0.160806 |