Calvin and Dan are playing a game of chance. Calvin tosses 8 fair coins, and wins if he obtains at least 4 heads. The probability that Calvin wins can be expressed as a/b where a,b are coprime positive integers. What is the value of

Question

a+b?

Answer 1

Use the symmetry w.r.t. interchanging heads and tails.

P(4 or more heads) = P(4 or more tails) = P(less than 5 heads).

So:

2 P(4 or more heads) = P(4 or more heads) + P(less than 4 heads) +
P(4 heads) = 1 + P(4 heads) ------>

P(4 or more heads) =

1/2 + 1/2 P(4 heads) =

1/2 + 1/2^9 Binomial(8,4) =

1/2 + 1/2^9 8*7*6*5/(4*3*2) =

1/2 + 35/2^8 =

(2^7 + 35)/2^8 =

163/256

Answer 2

a+b=419