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How do I calculate probabilities for a large number of independent events without using a tree. For example if toss a coin a 1000 times how do I calculate that it will be heads only 200 times?
Wiki User
∙ 11y ago
If p is the probability of the event happening (once), so that q = 1 - p is the probability of the event not happening (once), then the binomial theorem can be used:
(p + q)n = nC0 p0qn + nC1 p1qn-1 + ... + nCr prqn-r + ... + nCn-1 pn-1q1 + nCn pnq0
Where n is the number of trials and r is the number of successes required, and nCr is the number of combinations of r things from a set of n and is:
nCr = n!/(n-r)!r!
where n! is n factorial which is n x (n-1) x (n-2) x ... x 2 x 1. 0! is defined to be 1.
So assuming a normal, unbiased coin with p = pr(head) = 1/2 → q = pr(not_head) = 1 - 1/2 = 1/2, then the probability of getting (r =) 200 heads out of (n =) 1000 trials is:
nCr prqn-r = 1000C200 (1/2)200(1/2)800
= 1000!/(800! x 200!) (1/2)1000
Wiki User
∙ 11y ago
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