Let’s take an idea – we’ll call it a “pretty good” idea. Suppose that this “pretty good” idea would, given everyone’s votes, receive 300 votes up and 200 votes down, for a total score of +100. This means that 60% of people voted up, and 40% of people voted down. (Pbum votes have no effect on the score, or the data, and are therefore not counted.)

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Then, bumuming that the order in which the voters vote to be completely random, there is an (0.4)^6 = 0.004096 chance that the first six votes will all be down, and the entry will be removed.

p = 0.004096

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Furthermore, there is a chance that the idea will receive one up vote before being knocked down to -5 and taken out of consideration. This chance = (1 choose 6) * (0.6)(0.4)^7, where (1 choose 6) = 6: that is, there are six possible positions for the up vote, as the first, second, third, fourth, fifth, or sixth vote. It cannot be the 7th or the 8th vote.

p = 0.00589824

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Then, the chance of two positive votes and eight negative votes is:

p = ((2 choose 6)+6) * (0.6)^2 * (0.4)^8 = 21 * (0.6)^2 * (0.4)^8 = 0.0049545216

*Note that the 2 choose 6 accounts for the number of cases where both positive votes occur before the first 6 negative votes, and the +6 accounts for the cases where one of the positive votes occurs just before the seventh negative vote.

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Similarly, one may continue onwards for increasing exponents, and one may also adjust the values of the probabilities of a positive or negative vote. Using these numbers, we obtain a final probability of being downvoted before getting 4 positive votes = 0.0049545216 + 0.00589824 + 0.004096 = 0.0149487616, or approximately 1.5%

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Thus, we can clearly see that this “pretty good” idea, which I feel is reasonably deserving of not being immediately downvoted, has a 1.5% chance of not receiving even 4 positive votes. (I’d run the numbers, but I need to go atm) There is, of course, an even higher chance of it not receiving 10 positive votes. And thus I must ask: Does this indicate that an unacceptable proportion of “pretty good” ideas are not receiving fair consideration? The answer, of course, will depend on what the reader considers to be fair consideration. Still, there can be no doubt that reducing this proportion will increase the number of quality ideas being given fair consideration.

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Note that while pbum votes affect the % of voters that vote up or down, it has no impact upon these numbers. One must consider the same series of votes with 1 pbum vote, with 2 pbum votes, with 3 pbum votes, etc, which sums back into exactly the percentage shown above.