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

Given : X = number of heads
p = probability of getting head
`therefore p 1/2 `
` therefore q =1 -p = 1-1/2 =1/2 `
Given , n= 8
` therefore X ~ B( 8,1/2)`
The p.m f of X is given as
`P( X=x) = p p(x) = ""^n C_xP^xq^(n-x) `
i.e ` P(x) = ""^8 C_x(1/2 )^x ( 1/2 ) ^(8-5)`
`= ""^8 C_ 3(1/2)^5 (1/2)^3` ltbr gt `= (8xx 7 xx 6)/(1xx 2 xx 3) xx (1)/(256)= 7/32`
` therefore P [ X = 5] = 0.21875 `
Hence , the probability of getting exactly 5 heads is 0.21875.
(2) P ( getting heads at lest once )
`= [ X ge 1]=1 -P [x=0] `
` 1-p(0) =1 - ""^8C_0(1/2)^0(1/2)^(8-0)`
`1-(1/2)^8=1 -1/256 = 2555/256 `
` therefore p [ X ge 1] = 0.996 `
Hence , the probability of getting heads at least once is 0.996.