Suppose that an unfair coin comes up heads 527 of the time T

Suppose that an unfair coin comes up heads 52.7% of the time. The coin is flipped a total of 18 times. What is the probability that you get exactly 8 tails? What is the probability that you got at most 16 heads?

Solution

Using binomial distribution for probability:

nCr(p^r)(q^(n-r))

n = 18 ; r = 8; probability of heads p = 0.527 ; probablity of tails q = (1-p ) = 0.473

i) Probability of exactly 8 tails= 18C8(q^8)(p^10)

= 18C8(0.473)^8(0.527)^10

= [18!/(8!*10!)]((0.473)^8(0.527)^10

= 43758*(0.473)^8(0.527)^10

= 0.181

b) Probalility of getting at most heads:

= probalility of getting 0 heads upto 16 heads

So, Probalility of getting at most heads: = 1- (probability of 17 heads + probability of 18 heads)

= 1 - (18C17(0.527)^17(0.473)^1 + 18C18(0.527)^18 )

= 1- (0.000158+ 0.00000983 )

= 0.99983