SOLUTION: A fair coin is tossed repeatedly until a head is obtained. The probability that the coin has to be tossed at least four times is...........................................? [Give

1-P(H)-P(TH)-P(TTH) = 1-0.5-0.5²-0.5³ = 1-0.5-0.25-0.125 = 0.125

Edwin

The problem asks to find the probability to have a head UNDER THE CONDITION that the three first tosses give a tail.


It is the sum of probabilities

    P = P(4) + P(5) + P(6) + . . . 


where P(4) is the probability to get H first time at the 4-th toss;

      P(5) is the probability to get H first time at the 5-th toss, assuming that the outcome was T at the 4-th toss;

      P(6) is the probability to get H first time at the 6-th toss, assuming that the outcome was T at the 4-th and 5-th toss,

      and so on . . . (infinite sum)


Notice that  P(4) = P(TTTH) = ;  

             P(5) = .P(TTTTH) =  = ;

             P(6) = .P(TTTTTH) =  =  ,   and so on . . . 


So,  P is the sum of the infinite geometric progression with the first term    and the common ratio of  = .

The sum of this progression is  P =  =  =  =  = 0.083333...   


It is the ANSWER to this problem.