# Ask The Wizard #237

### Probability of MatchingNumbers in 6/49 Lottery

Years | Probability |

5 | 0.009640 |

10 | 0.038115 |

15 | 0.083800 |

20 | 0.144158 |

25 | 0.215822 |

30 | 0.295459 |

35 | 0.379225 |

40 | 0.463590 |

45 | 0.545437 |

50 | 0.622090 |

55 | 0.691985 |

60 | 0.753800 |

65 | 0.807008 |

70 | 0.851638 |

75 | 0.888086 |

80 | 0.917254 |

85 | 0.940000 |

90 | 0.957334 |

95 | 0.970225 |

100 | 0.971954 |

In case you were wondering, the number of draws for the probability of a matching draw to first exceed 50% is 4,404.

So, assuming 52% of resolved bets win, the overall probabilities are:

Win: 50.44%

Draw: 3.00%

Loss: 46.56%

Using basic statistics, it is easy to see that the expected win per pick, laying -110, is -0.0078. The standard deviation per pick is 1.0333. The expected win over 70 picks is -0.5432, and the standard deviation is 70^{1/2}×1.0333 = 8.6452. A win of 8.5 units is 9.0432 units above expectations, or 9.0432/8.6452=1.0460 standard deviations to the right of expectations on the Gaussian Curve. I think we can ignore the adjustment for a discrete distribution because of the pushes, and some games not being -110/-110, will result in a fairly smooth curve down the a factor of 0.05 units.

So, the probability of any one player finishing more than 1.046 standard deviations above expectations is 14.77%. That figure can be found in any table of the Gaussian curve, or with the formula =1-normsdist(1.046) in Excel. The probability of all six players finishing under 1.046 is (1-0.1477)^{6}=38.31%. Thus, the probability of at least one player finishing above 1.046 standard deviations up is 61.69%. That makes the over look like a solid bet laying -110. I show it is fair at -161.

The following table shows the probability of the over 8.5 winning given various values of p. Perhaps the person setting the prop was assuming a value closer to 51% for p.

### NFL Handicapping Prop

Prob. Correct Pick | Prob. Over Wins |

50.0% | 41.16% |

50.5% | 46.18% |

51.0% | 51.33% |

51.5% | 56.53% |

52.0% | 61.69% |

52.5% | 66.72% |

53.0% | 71.52% |