Question

A biased coin is tossed till a head appears for the first time . What is the probability that the number of required tossed is odd.
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Answer 1

Answer:

The coin is tossed until the head appears. Let p be the probabilty of getting a head and q=1−p be the probabilty of getting a tail. Probabilty that required number of tosses is odd means that the head appears in first trial or third trial or fifth trial and so on. Now, the probabilty that the head appears in first trial is p. Head appears in 3rd trial is q∗q∗p=q2p, it appears in 5th trial is q∗q∗q∗q∗p=q4p....

Hence, the required probabilty is p+q2p+q4p+q6p+.....=p(1+q2+q4+q6+...)=p(1−q2)−1=p1−q2=p(1+q)(1−q)=11+q