Question

In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is . What is the probability that he will knock down fewer than 2 hurdles?

Answer 1

Let p and q respectively be the probabilities of that player who will clear and knock down the hurdle.

Probability of clearing each hurdle is given by,

p= 5 6

Probability of knocking down each hurdle is given by,

q=1− 5 6 = 1 6

Let X be the random variable representing the number of times the player will knock down the hurdle.

The formula for the binomial distribution is given by,

p( X=x )= C n a p n−x q x

P (player knocking down the two hurdles) =P( X<2 )

P( X<2 )=P( X=0 )+=P( X=1 ) = C 10 0 p 10−0 q 0 + C 10 1 p 10−1 q 1

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

P( X<2 )=1× ( 5 6 ) 10 ( 1 6 ) 0 +10× ( 5 6 ) 9 ( 1 6 ) 1 = ( 5 6 ) 9 [ 5 6 + 10 6 ] = ( 5 6 ) 9 ( 5 2 ) = 5 10 2× 6 9

Thus, the probability of knocking down fewer than two hurdles is 5 10 2× 6 9 .