Question

A) A basketball player is an 80% foul shooter. Find the probability of her making:

1. 7 of 10 foul shots

2. 8 of 10 foul shots

3. 10 for 10 foul shots

B) Over several games the shooter is given 100 foul shots. Find (i.e. estimate):

1. She makes more than 80 shots

2. She makes less than 80 shots

3. She makes less than 50 shots

Please show all work and graphs

Answer 1

Question A)

X ~ B ( n = 10 , P = 0.8 )

Part 1

Part 2

Part 3

Question B)

Using Normal Approximation to Binomial
Mean = n * P = ( 100 * 0.8 ) = 80
Variance = n * P * Q = ( 100 * 0.8 * 0.2 ) = 16
Standard deviation = √(variance) = √(16) = 4

Part 1

P ( X > 80 )
Using continuity correction
P ( X > n + 0.5 ) = P ( X > 80 + 0.5 ) = P ( X > 80.5 )

X ~ N ( µ = 80 , σ = 4 )
P ( X > 80.5 ) = 1 - P ( X < 80.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 80.5 - 80 ) / 4
Z = 0.13
P ( ( X - µ ) / σ ) > ( 80.5 - 80 ) / 4 )
P ( Z > 0.13 )
P ( X > 80.5 ) = 1 - P ( Z < 0.13 )
P ( X > 80.5 ) = 1 - 0.5517
P ( X > 80.5 ) = 0.4483

Part 2

P ( X < 80 )
Using continuity correction
P ( X < n - 0.5 ) = P ( X < 80 - 0.5 ) = P ( X < 79.5 )

X ~ N ( µ = 80 , σ = 4 )
P ( X < 79.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 79.5 - 80 ) / 4
Z = -0.13
P ( ( X - µ ) / σ ) < ( 79.5 - 80 ) / 4 )
P ( X < 79.5 ) = P ( Z < -0.13 )
P ( X < 79.5 ) = 0.4483

Part 3

P ( X < 50 )
Using continuity correction
P ( X < n - 0.5 ) = P ( X < 50 - 0.5 ) = P ( X < 49.5 )

X ~ N ( µ = 80 , σ = 4 )
P ( X < 49.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 49.5 - 80 ) / 4
Z = -7.63
P ( ( X - µ ) / σ ) < ( 49.5 - 80 ) / 4 )
P ( X < 49.5 ) = P ( Z < -7.63 )
P ( X < 49.5 ) = 0