Question

A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

Answer 1

Let p and q respectively be the probabilities of getting and not getting the third six in the sixth throw of die respectively.

Probability of getting a six in a throw is given by,

p= 1 6

Probability of not getting a six in a throw is given by,

q=1− 1 6 = 5 6

Let X be the random variable representing the number of times the player is not getting a six in a throw.

The formula for the binomial distribution is given by,

p( X=x )= C n a p n−x q x

P (2 sixes comes in the first five throws of die) =P( X=2 )

P( X=2 )= C 5 2 p 5−2 q 2

Substitute the values of p and q in the above expression.

P( X=2 )= C 5 2 ( 1 6 ) 5−2 ( 5 6 ) 2 (1)

Probability of getting a third six in sixth throw.

= 10× ( 5 ) 3 ( 6 ) 5 × 1 6 = 10×125 ( 6 ) 6 = 10×125 46656 = 625 23328

Thus, the probability of getting third six in the sixth throw is 625 23328 .