Question

1. A dart player throws 10 darts at the dart board. He has a 20% chance of hitting the bull’s eye each time. Suppose throws are independent.
   1. What is the probability that he hits the bull’s eye 4 times?
   2. What is the probability that he hits the bull’s eye at least twice?
   3. How many times would he be expected to hit the bull’s eye?
   4. What is the standard deviation of the number of times he is likely to hit the bulls eye?

Answer 1

X ~ Bin ( n, p )

Where n = 10 , p = 0.20

Binomial probability distribution is

P(X) = ⁿC_X * p^X * ( 1 - p)^n-X

a)

P(X = 4) = ¹⁰C₄ * 0.20⁴ * ( 1 - 0.20)⁶

= 0.0881

b)

P(X >= 2) = 1 - P(X <= 1)

= 1 - [ P(X = 0) + P(X = 1) ]

= 1 - [ ¹⁰C₀ * 0.20⁰ * ( 1 - 0.20)¹⁰ +¹⁰C₁ * 0.20¹ * ( 1 - 0.20)⁹ ]

= 0.6242

c)

E(X) = n * p

= 10 * 0.20

= 2

d)

Standard deviation = sqrt [ n p ( 1 - p) ]

= sqrt [ 10 * 0.20 ( 1 - 0.20) ]

= 1.2649