Question

(9) Shaquille O’Neal is practicing his free throws in the gym. Shaq’s probability of making a free throw over his career is 0.527. He will shoot 150 free throws. a) Define a random variable, and write out the probability mass function for the number of free throws Shaq makes on his 150 attempts. b) What is the probability that Shaq makes between 78 and 80 free throws, inclusive? c) What is the expected value and variance of the number of free throws Shaq will make during his practice

Answer 1

Solution

Back-up Theory

If an experiment is dichotomous, i.e., has only two possible outcomes, say success and failure, which are mutually exclusive and collectively exhaustive, n repetitions of the experiment are identical, outcomes are independent and probability of a success, say p, is constant, then the random variable X, that represents the number of successes of the experiment, is a binomial variable and it is represented as: X ~ B(n, p) ………………………………………………..................................…….. (1)

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and p = probability of one success, then, probability mass function (pmf) of X is given by p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}.............…...........……..(2)

[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST]..............................................(2a)

Mean (average) of X = E(X) = µ = np….....................................................................……………………………………………..(3)

Variance of X = V(X) = σ² = np(1 – p)………….................................................................………………………………………..(4)

Standard Deviation of X = SD(X) = σ = √{np(1 – p)} ……......................................................……………………..……………...(5)

Normal approximation

X ~ B(n, p) np ≥ 5 and np(1 - p) ≥ 5, (X – np)/√{np(1 - p)} ~ N(0, 1) [approximately] ............................................................. (6)

If a random variable X ~ N(µ, σ²), i.e., X has Normal Distribution with mean µ and variance σ²,

then, Z = (X - µ)/σ ~ N(0, 1), i.e., Standard Normal Distribution and hence

P(X ≤ or ≥ t) = P[{(X - µ)/σ} ≤ or ≥ {(t - µ)/σ}] = P[Z ≤ or ≥ {(t - µ)/σ}] .………...........................................................………...…(7)

Probability values for the Standard Normal Variable, Z, can be directly read off from Standard Normal Tables.................... (7a)

or can be found using Excel Function: Statistical, NORMSDIST(z) which gives P(Z ≤ z) ..................................................…(7b)

Now to work out the solution,

Let X = the number of free throws Shaq makes on his 150 attempts.

Part (a)

Vide (1),

X ~ B(150, 0.527) ................................................................................................................................................................... (8)

So, vide (8) and (2),

the probability mass function for the number of free throws Shaq makes on his 150 attempts is: p(x) = P(X = x) = (¹⁵⁰C_x)(0.527^x)(0.473)^{150 – x} Answer 1

Part (b)

Probability that Shaq makes between 78 and 80 free throws, inclusive

= P(78 ≤ X ≤ 80)

= P[{(78 – 79.05)/√{37.39065)} ≤ {(X – np)/√{np(1 - p)} ≤ {(80 – 79.05)/√{37.39065)}] [vide (6)]

= P(- 0.1718 ≤ Z ≤ 0.1555)

= P(Z ≤ 0.1555) - P(Z ≤ - 0.1718)

= 0.5618 – 0.4318 [vide (7b)]

= 0.13 Answer 2

Part (c)

Vide (3),

Expected value of the number of free throws Shaq will make during his practice

= 150 x 0.572 = 79.05 Answer 3

Vide (4),

Variance of the number of free throws Shaq will make during his practice

= 150 x 0.542 x 0.458

= 37.3907 Answer 4

DONE