Example 31 If a fair coin  is tossed 10 times, find the probability of
(i)
exactly six heads
(ii)
at least six heads
(iii)
at most six heads
Solution The repeated tosses of a coin are Bernoulli trials Let X denote the number
of heads in an experiment of 10 trials Clearly, X has the binomial distribution with n = 10 and p = 1
2
Therefore
P(X = x) = nCxqn–xpx, x = 0, 1, 2, ,n
Here
n = 10, 
21
p
 , q = 1 – p = 1
2
Therefore
P(X = x) =
10
10
10
10
1
1
1
C
C
2
2
2
x
x
x
x
−
⎛
⎞
⎛
⎞
⎛
⎞
=
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
Now
(i) P(X = 6) =
10
10
6
10
1
10