How to figure how unlikely a certain outcome of a 50% event would be? EG, flipping 700 heads out of 1000 flips

Answer 1

Problems of that nature are solved with a binomial distribution. The probability of x "successes" in y "attempts" (where probability of success is z (0<=z<=1)), is:

(z^x) * ((1-z)^(y-x)) * (y! / (x! (y-x)!)

... that's the probability of x successes, times the probability of y-x failures, times the number of combinations for that specific set of outcomes in y trials.

In the case of x=700 (heads), y=1000 (flips), and z=0.5 (odds of flipping heads), it's:

(z^x) * ((1-z)^(y-x)) * (y! / (x! (y-x)!)
(0.5^700) * (0.5^300) * (1000! / (700! * 300!))

Which is approximately:
2x10^-211 * 5x10^-91 * (4x10^2567 / (2x10^1689 * 3x10^614))

... which works out to about 5 x 10^-38. That's about one in 20, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000.

Answer 2

The unlikelihood of an event is 1 minus the likelihood of the event. For example, if there were three balls in a bag (red, yellow, blue), the likelihood of drawing the red ball is 1 in 3 or 33% or 0.33. The likelihood of not drawing out the red ball is 2 in 3 or 66% or 0.66...1 - 0.33.