United States
Member #197,030
March 28, 2019
1,647 Posts
Offline
For this post, the "span" of a combination is the difference between the largest and smallest numbers drawn. E.g., the Fantasy Five ticket 2-11-21-24-29 has a span of 27. In 5/N and 6/N style games, for a given N, I found a number K such that about 50% of the time (or as close to 50% as I can get) the span is at most K.
These findings are given below.
5/N games "span" is max - min N = 32: 49.402% chance span is at most 22 N = 33: 51.488% chance span is at most 23 N = 34: 47.098% chance span is at most 23 N = 35: 49.099% chance span is at most 24 N = 36: 51.004% chance span is at most 25 N = 37: 52.818% chance span is at most 26 N = 38: 48.846% chance span is at most 26 N = 39: 50.599% chance span is at most 27 N = 40: 52.276% chance span is at most 28 N = 41: 48.633% chance span is at most 28 N = 42: 50.257% chance span is at most 29 N = 43: 51.815% chance span is at most 30 N = 44: 48.45% chance span is at most 30 N = 45: 49.962% chance span is at most 31 N = 46: 51.418% chance span is at most 32 N = 47: 48.292% chance span is at most 32 N = 48: 49.707% chance span is at most 33 N = 49: 51.073% chance span is at most 34 N = 50: 48.154% chance span is at most 34 N = 51: 49.483% chance span is at most 35 N = 52: 50.769% chance span is at most 36 N = 53: 48.032% chance span is at most 36 N = 54: 49.286% chance span is at most 37 N = 55: 50.501% chance span is at most 38 N = 56: 51.679% chance span is at most 39 N = 57: 49.11% chance span is at most 39 N = 58: 50.261% chance span is at most 40 N = 59: 51.38% chance span is at most 41 N = 60: 48.952% chance span is at most 41 N = 61: 50.046% chance span is at most 42 N = 62: 51.111% chance span is at most 43 N = 63: 48.81% chance span is at most 43 N = 64: 49.853% chance span is at most 44 N = 65: 50.868% chance span is at most 45 N = 66: 48.682% chance span is at most 45 N = 67: 49.677% chance span is at most 46 N = 68: 50.648% chance span is at most 47 N = 69: 48.565% chance span is at most 47 N = 70: 49.517% chance span is at most 48
6/N games "span" is max - min N = 36: 52.499% chance span is at most 27 N = 37: 47.459% chance span is at most 27 N = 38: 49.247% chance span is at most 28 N = 39: 50.958% chance span is at most 29 N = 40: 52.596% chance span is at most 30 N = 41: 48.068% chance span is at most 30 N = 42: 49.665% chance span is at most 31 N = 43: 51.199% chance span is at most 32 N = 44: 52.674% chance span is at most 33 N = 45: 48.564% chance span is at most 33 N = 46: 50.006% chance span is at most 34 N = 47: 51.397% chance span is at most 35 N = 48: 47.617% chance span is at most 35 N = 49: 48.976% chance span is at most 36 N = 50: 50.29% chance span is at most 37 N = 51: 51.561% chance span is at most 38 N = 52: 48.078% chance span is at most 38 N = 53: 49.323% chance span is at most 39 N = 54: 50.53% chance span is at most 40 N = 55: 51.701% chance span is at most 41 N = 56: 48.47% chance span is at most 41 N = 57: 49.619% chance span is at most 42 N = 58: 50.736% chance span is at most 43 N = 59: 51.821% chance span is at most 44 N = 60: 48.808% chance span is at most 44
For example, in CA Fantasy Five, which is a 5/39 game, about 50.599% of the combinations have a span of 27 or less. In GA Jumbo Bucks Lotto, which is a 6/47 game, about 51.397% of the combos have a span of 35 or less.
United States
Member #35,334
March 16, 2006
238 Posts
Offline
With 974 Cash4Life draws since the last format change for real world data, a 5/60-4 game, the 50th percentile of the drawn numbers span (max - min, not the extra ball) is 41.
If one were to calculate the number of times a ball appears in positions 1 thru 5 for every possible combination, exclude the range that most closely gets to the 50th percentile, the span is 44.
United States
Member #197,030
March 28, 2019
1,647 Posts
Offline
Quote: Originally posted by GoogilyMoogily on Dec 22, 2020
With 974 Cash4Life draws since the last format change for real world data, a 5/60-4 game, the 50th percentile of the drawn numbers span (max - min, not the extra ball) is 41.
If one were to calculate the number of times a ball appears in positions 1 thru 5 for every possible combination, exclude the range that most closely gets to the 50th percentile, the span is 44.
How did you calculate your span?
Here's how I found the 50-ish percentile. Let's take the 5/N game and fix N=60 like your Cash4Life example, ignoring the extra ball. There are 60 choose 5 = 5461512 different ticket combinations ignoring the extra ball. Half of 5461512 is 2730756.
In a 5/60 game, the span of a combo can be as small as 4, or as large as 59. What I did first is find the number of combos for each span from 4 to 59:
span of 4: 56 combinations
span of 5: 220 combinations
span of 6: 540 combinations
span of 7: 1060 combinations
span of 8: 1820 combinations
span of 9: 2856 combinations
span of 10: 4200 combinations
span of 11: 5880 combinations
span of 12: 7920 combinations
span of 13: 10340 combinations
span of 14: 13156 combinations
span of 15: 16380 combinations
span of 16: 20020 combinations
span of 17: 24080 combinations
span of 18: 28560 combinations
span of 19: 33456 combinations
span of 20: 38760 combinations
span of 21: 44460 combinations
span of 22: 50540 combinations
span of 23: 56980 combinations
span of 24: 63756 combinations
span of 25: 70840 combinations
span of 26: 78200 combinations
span of 27: 85800 combinations
span of 28: 93600 combinations
span of 29: 101556 combinations
span of 30: 109620 combinations
span of 31: 117740 combinations
span of 32: 125860 combinations
span of 33: 133920 combinations
span of 34: 141856 combinations
span of 35: 149600 combinations
span of 36: 157080 combinations
span of 37: 164220 combinations
span of 38: 170940 combinations
span of 39: 177156 combinations
span of 40: 182780 combinations
span of 41: 187720 combinations
span of 42: 191880 combinations
span of 43: 195160 combinations
span of 44: 197456 combinations
span of 45: 198660 combinations
span of 46: 198660 combinations
span of 47: 197340 combinations
span of 48: 194580 combinations
span of 49: 190256 combinations
span of 50: 184240 combinations
span of 51: 176400 combinations
span of 52: 166600 combinations
span of 53: 154700 combinations
span of 54: 140556 combinations
span of 55: 124020 combinations
span of 56: 104940 combinations
span of 57: 83160 combinations
span of 58: 58520 combinations
span of 59: 30856 combinations
You can check that if you add up all the numbers of combinations you get 5461512.
Now, let's add these numbers of combinations starting at span=4 going all the way to span=K, taking K to be the number such that this cumulative sum is as close to 2730756 as possible. In this example, that value of K turns out to be 41. The exact sum from span=4 to span=41 is 2673528, and 2673528/5461512 is approximately 48.952%. That's as close to 50% as I can get with these parameters. (If I summed up to span=42, I would get a sum of 2865408, and 2865408/5461512 is approximately 52.465%, which is not as close to 50%.)