Edit question According to ESPN, LeBron James has a regular season free-throw su

Question

Edit question According to ESPN, LeBron James has a regular season free-throw success rate of 75% over his playing career to date. assume that the free-throw shots are independent, that is, success or failure on one shot does not affect the chance of success on another shot. use this information to answer the next two questions:

A) Consider 10 free-throws taken by LeBron James, what is the probability that at least 8 of them are successful?

B) Consider 100 free-throws taken by LeBron James, what is the probability that at least 80 of them are successful? Use the normal approximation if appropriate.

Explanation / Answer

Let X be the random variable that number of successful free throws.

A) Consider 10 free-throws taken by LeBron James, what is the probability that at least 8 of them are successful?

X ~ Binomial(n = 10, p = 0.75)

The probability mass function of binomial distibution is,

P(X=x) = (n C x)*px*(1-p)n-x

And we have to find P(X>=8).

P(X>=8) = P(X=8) + P(X=9) + P(X=10)

= (10 C 8)*0.758 * 0.252 + (10 C 9)*0.759 * 0.251 + (10 C 10)*0.7510*0.250

= 0.2816 + 0.1877 + 0.0563

P(X >=8) = 0.5256

B) Consider 100 free-throws taken by LeBron James, what is the probability that at least 80 of them are successful? Use the normal approximation if appropriate.

Here n=100 and p = 0.75

If np >=10 and n(1-p) >=10 then we go for normal distribution.

np = 100*0.75 = 75

n(1-p) = 100*(1-0.75) = 25

Both are greator than 10 so we use here normal approximation.

mean = np = 75

var = n*p*(1-p) = 100*0.75*0.25 = 18.75

sd = sqrt(var) = sqrt(18.75) = 4.3301

Now we have to find P(X >=80) = 1 - P(X <80)

Convert x=80 into z-score.

z = (x - mean) / sd

z = (80 - 75) / 4.3301 = 1.1547

That is now we have to find P(X>=1.1547).

P(Z >=1.1547) = 1 - P(Z < 1.1547)

This probability we can find by using EXCEL.

syntax is,

=NORMSDIST(z) (EXCEL always gives left tail probability)

where z is test statistic value = 1.1547

P(Z >=1.1547) = 1 - P(Z < 1.1547)

= 1 - 0.8759 = 0.1241

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