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) = ¹⁰⁰C_x (1/2)^x (1/2)^100-x = ¹⁰⁰C_x (1/2)¹⁰⁰

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

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

= ¹⁰⁰C₄₁ (1/2)¹⁰⁰ + ¹⁰⁰C₄₂ (1/2)¹⁰⁰ + ... + ¹⁰⁰C₅₉ (1/2)¹⁰⁰

= (1/2)¹⁰⁰ [¹⁰⁰C₄₁ + ¹⁰⁰C₄₂ +... + ¹⁰⁰C₅₉]

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