Consider an experiment where a fair coin is tossed until a h

Consider an experiment where a fair coin is tossed until a head comes up the first time. Let X denote the number of tosses until the first head appears, find P(X>=2) directly and using complement rule.

Solution

P(X>=2) = P(X<2)

= 1 - [ P(X = 0) + P(X = 1) ]

= 1 - [0 + (1/2) ]

= 1 - 1/2

= 1/2

Probability = 0.5

Direct moethod:

P(X>=2) = {(TH),(TTH),(TTTH)........} = (1/2)(1/2) + (1/2)(1/2)(1/2) + (1/2)(1/2)(1/2)(1/2) + (1/2)(1/2)(1/2)(1/2)(1/2) +...........

= 1/4 + 1/8 + 1/16 + 1/ 32 ..........

= [(1/4) / {1 - (1/2) }] = (1/4) / (1/2) = 1/2