Using the HGD provides a close estimate. For the true probabilities, you need to consider that not all distributions of X calls are equal, and the distribution is something that all cards in play are subject to.
So you have to split the calculation into two parts. Let's say you want the probability of a coverall in X calls (or less) when Y cards are in play:
1) Identify (iterate through) every possible distribution of X calls. For example, with 50 calls, there are 20,176 possible distributions (including distributions where a coverall is not possible). Determine the probability of each distribution.
Then, for each distribution:
2) Determine the probability of achieving a coverall given the current distribution. This will of course be zero when there are fewer than 4 Ns or 5 Bs/Is/Gs/Os.
3) Apply the 1-((1-p)Y
) formula here with p = the result of #2 and Y = the number of cards in play.
4) Multiply the result of #3 by the distribution probability of #1, and accumulate (sum) these results for each distribution to determine the overall probability of a bingo within X calls or less for Y cards.
For example, for a coverall in 50 calls, here are the true probabilities (vs. the HGD estimates) for a few card quantities:
1 card = 0.0000047150811362 (vs. the HGD estimate of 0.0000047150811337)
100 cards = 0.00047138 (vs. the HGD estimate of 0.00047140)
500 cards = 0.00235428 (vs. the HGD estimate of 0.00235477)
1000 cards = 0.00470206 (vs. the HGD estimate of 0.00470399)
5000 cards = 0.02325221 (vs. the HGD estimate of 0.02329973)
The above calculations assume the following:
- standard North American bingo cards are used (5 Bs, 5 Is, 4 Ns, 5 Gs, 5 Os)
- the cards in play are random
- the numbers are drawn randomly
Last edited by: JB on Mar 2, 2017