A fair coin is tossed 8 times. Find the probability that it shows heads
(i) exactly 5 times (March ' 17)
(2) at lest once . (March ' 14- 17 )

Explanation

Let X= no. of heads in 8 trials
p= Probability of head up in a trial
`:. P=(1)/(2),q=1-(1)/(2)=(1)/(2)`
Here n=8
`:. X~B(n,p)`
`:. X~B(8,1//2)`
The p.m.f of X is given by
`P(X=x) = ""^(n)C_(x)P^(x)8a^(n-x)`
`P(X=x)= ""^(8)C_(x)*(1//2)^(8-5)....(1)`
(i) Probability that is shows exactly 5 times heads. .
Here,x=5
`:. P(X=5)=""^(8)C_(5)xx(1//2)^(5)xx(1//2)^(8-5)`
`=(8!)/(5!(8-5)!)xx(1//2)^(5)xx(1//2)^(3)`
`(8xx7xx6xx5!)/(5!xx3xx2xx1)(1//2)^(8)`
`=56xx(1)/(26)=(7)/(32)=0*21187`
(ii) Probability that it show head at least once.
`xge1`
`:. P(X ge1)= 1-P(X lt1)`
`=1-P(X=0)`
`=1-""^(8)C_(0)(1//2)^(0)(1//2)^(8-0)`
`=1-1xx1(1//2)^(8)`
`=1-(1)/(256)=(255)/(256)`
`0*99609`