You flip a fair coin 10 times What is the probability that i

You flip a fair coin 10 times. What is the probability that it lands on heads exactly 5 times? The probability of exactly 5 heads is 252/1024. What is the probability that it lands on heads at least 5 times? The probability of at least 5 heads is 2.

Solution

This is binomial experiment.

Probability of r success in n trials = P(X=r) = ⁿC_r p^r(1-p)^(n-r)

here n = 10 , p = 0.5 (fair coin) ,

a) r = 5

P(X=5) = 10C5 (0.5)^5(1-0.5)^5

252/1024

b) P(X>=5) = P(X=5)+ P(X=6) + ... P(X=10)

since p = 0.5 ,it is symmetric. P(X= r) = P(X = (n-r))

now let y =  P(X=6) + ... P(X=10)

and x = P(X=5)

we have to calculate x + y

we know that P(x=0) + P(X=1) + ... P(x=5) +  P(X=6) + ... P(X=10) = 1

or y+ x+ y = 1

x + 2y = 1

since x = 252/1024

x + y = 1/2 ( x+2y+ x)

= 1/2 (1+252/1024)

= 1/2 + 126/1024

= 638/1024