Agnes has a weighted coin so that heads comes up 23 of the t

Agnes has a weighted coin so that heads comes up 2/3 of the time. She tosses her coin five

times. What is the probability that heads will appear for the rst time on toss 5?

Solution

Probability of getting a head, p = 2/3

Probability of getting a tail, (1-p) = 1/3

Now,

P ( getting H at the 1^st draw ) = 2/3

P ( getting H at the 2^nd draw ) = P (getting T at the 1^st draw and H at the 2^nd draw ) = 1/3 x 2/3 = 2/9

P ( getting H at the 3^rd draw ) = P (getting T at the 1^st and 2^nd draw and H at the 3^rd draw ) = 1/3 x 1/3 x 2/3 = 2/27

P ( getting H at the 4^th draw ) = P (getting T at the 1^st , 2^nd and 3^rd draw and H at the 4^th draw ) = 1/3 x 1/3 x 1/3 x 2/3 = 2/81

P ( getting H at the 5^th draw ) = P (getting T at the 1^st , 2^nd , 3^rd and 4^th draw and H at the 5^th draw ) = 1/3 x 1/3 x 1/3 x 1/3 x 2/3 = 2/243