Question

Liam is a professional darts player who can throw a bullseye 70% of the time.

If he throws a dart 250 times, what is the probability he hits a bulls eye:

a.) At least 195 times?  

b.) No more than 175 times?  

c.) between 160 and 195 times (including 160 and 195)?  

Use the Normal Approximation to the Binomial distribution to answer this question.

Answer 1

Solution:

Given that,

P = 0.70

1 - P = 0.30

n = 250

Here, BIN ( n , P ) that is , BIN (250 , 0.70)

According to normal approximation binomial,

X Normal

Mean = = n*P = 175

Standard deviation = =n*p*(1-p) = 52.5

We using continuity correction factor

a)

P(X a ) = P(X > a - 0.5)

P(x > 194.5) = 1 - P(x < 194.5)

= 1 - P((x - ) / < (194.5 - 175) / 52.5)

= 1 - P(z < 2.69)

= 1 - 0.9964   

= 0.0036

Probability = 0.0036

b)

P( X a ) = P(X < a + 0.5)

P(x < 175.5) = P((x - ) / < (175.5 - 175) / 52.5)

= P(z < 0.069)

Probability = 0.5275

c)

P( a - 0.5 X b + 0.5) = P( a - 0.5 < X < b + 0.5)

P(159.5 < x < 195.5) = P((159.5 - 175)/ 52.5) < (x - ) /  < (195.5 - 175) / 52.5) )

= P(-2.14 < z < 2.83)

= P(z < 2.83) - P(z < -2.14)

= 0.9977 - 0.0162

Probability = 0.9815