JAG331 -
Math is right, conclusions are in error.
In a 52 ball pool there are 26 odd numbers and 26 even numbers. Odds of drawing an odd-numbered ball are 1::2.00.
There are 51 balls left, 25 odd, 26 even. Odds of drawing an odd-numbered ball are 25::51, or 1::2.04. Cumulative odds, those of drawing two odd-numbered balls from 52, are 1::4.08.
There are 50 balls left, 24 odd, 26 even. Odds of drawing an odd-numbered ball are 24::50, or 1::2.08. Cumulative odds, those of drawing three odd-numbered balls from 52, are 1::8.50.
There are 49 balls left, 23 odd, 26 even. Odds of drawing an odd-numbered ball are 23::49, or 1::2.13. Cumulative odds, those of drawing four odd-numbered balls from 52, are 1::18.11.
There are 48 balls left, 22 odd, 26 even. Odds of drawing an odd-numbered ball are 22::48, or 1::2.18. Cumulative odds, those of drawing five odd-numbered balls from 52, are 1::39.51.
Then, one ball is drawn from a separate pool of 52. Odds of drawing an odd-numbered ball are 26::52 or 1::2.00. Multiplying these odds together gives you a chance of drawing all odd (or all even) numbers at 1::79.02.
The proof is from another direction. If only odd numbers from 1 to 52 are considered, there are 26 numbers. If you were to have a 5/26 lottery matrix there would be 65,780 possibilities. In a 5/52 lottery matrix there are 2,598,960 possibilities. Dividing 2,598,960 by 65,780 gives you odds of drawing any 26 given numbers in a 5/52 lottery matrix, whether all odd, all even, 26 and under or 27 and over, of 1::39.51. Multiplying that by the 1::2.00 bonus ball odds yields game odds of 1::79.01.
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