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# If you flip a coin 100 times, what is the probability of getting between 40 and 60 heads?

**Solution: **

Clearly given is a random experiment with 100 Bernoulli trials with,

p(success) = p(Head) = 1/2 = p

p(Failure) = 1 - p = 1 - 1/2 = 1/2 = q

p(X = x) = nC_{x} p^{x} q^{n-x}

Here, n = 100

Thus by using the binomial distribution calculation, we get

p(X = x) = ^{100}C_{x} (1/2)^{x} (1/2)^{100-x} =^{ 100}C_{x} (1/2)^{100}

p(Getting heads between 40 and 60) = p(40 < x <60)

= p(x = 41) + p(x =42) + ... + p(x = 59)

= ^{100}C_{41} (1/2)^{100 }+ ^{100}C_{42} (1/2)^{100 }+ ... + ^{100}C_{59} (1/2)^{100}

= (1/2)^{100} [^{100}C_{41} + ^{100}C_{42} +... + ^{100}C_{59}]

p(40 < x <60) =0.9431

## If you flip a coin 100 times, what is the probability of getting between 40 and 60 heads?

**Summary:**

If you flip a coin 100 times, the probability of getting between 40 and 60 heads is 0.9431

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