Reply #70: Hope this clears it up.

[TruthIsAll]

These are the probabilities that EXACTLY N of 88 voting machines would randomly switch Kerry votes to Bush. There are actually 89 possibilities: N can equal 0, 1, 2...88. The sum of ALL 89 probabilities is equal to 1.

A vote switch, assuming it is RANDOM, has an equal 0.5 probability of favoring either Bush or Kerry.

We need to calculate the probability P that AT LEAST 86 of 88 machines would flip to Bush. That is equivalent to the calculating the SUM of the probabilities for the three cases where N = 86,87,88.

In Excel, we use the Binomial Distribution function to determine the chance that exactly X of N events would occur, given that p is the probability of any single occurrence.

1)To determine the probability that EXACTLY X occur:
p (X) = BINOMDIST(X, N, p, FALSE)
FALSE is a flag to calculate the exact probability.

Another method:
2) To determine the probability that AT LEAST X occur:
P (>=X) = 1 - BINOMDIST(X-1, N, p, TRUE)
Note: P = 1 - probability that at MOST X-1 occur.

TRUE is a flag to calculate the cumulative probability.

Using method 1:
Solving for each of the three possibilities:
p (86) = BINOMDIST(86, 88, .5, FALSE)
p (87) = BINOMDIST(87, 88, .5, FALSE)
p (88) = BINOMDIST(88, 88, .5, FALSE)

The probability is:
P = p(86)+ p(87) + p(88) or 1 in 79 sextillion

Using method 2 (cumulative):
P (>=X) = 1 - BINOMDIST(85, 88, 0.5, TRUE)

Unfortunately, Excel cannot calculate the probability using the cumulative option. It's too small.