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 701/2×1.0333 = 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% |