Details

Snapshot 1: The figure shows the probabilities that the bill is defeated with 100 voters, of whom 49 are against the bill, 51 are for the bill, and 0, 1, 2, …, 100 voters are missing. For example, if 3 voters are missing, the probability is 0.128788; if 4 voters are missing, the probability is 0.0637301; and if 5 voters are missing, the probability is 0.193846. If 0, 1, or 2 voters are missing, the bill will never be defeated. For example, if 1 voter is missing, that voter is, in the worst case, for the bill, so that we have 49 against and 50 for the bill. If 2 voters are missing, they are, in the worst case, both for the bill, so that we have 49 against and 49 for; in this case the bill is accepted.

Snapshot 2: Now we have 50 voters against the bill and 50 for it. If an odd number of voters is missing, the probability that the bill is defeated is 0.5. For example, if 1 voter is missing, that voter is equally likely to be for as against the bill.

Snapshot 3: We have 51 against and 49 for the bill. The probability that the bill is defeated is 1 if 0 or 1 voter is missing.

Recall that we have a total of voters, of whom voters are against the bill and voters are for the bill. Also, voters are missing. Let be the random variable that is the number of voters missing the vote who, if present, would have voted for the bill. Then we have voters missing the vote who would have voted against the bill. Therefore, we have voters for the bill and against the bill. The bill will be defeated if or . This condition can also be written as , where is used for the floor function (the greatest integer less than or equal to the argument).

The random variable has the hypergeometric distribution. In Mathematica, this distribution has three parameters , , and . The hypergeometric distribution gives the distribution of the number of successes in draws from a population of size containing successes. Now we have , , and . Thus,

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The probability that the bill is defeated is then

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The bill is accepted if there are as many votes for the bill as there are against it.

The Demonstration is based on problem 11 in [1].

Reference

[1] P. J. Nahin, Digital Dice: Computational Solutions to Practical Probability Problems, Princeton: Princeton University Press, 2008.