Question

In a​ region, 90​% of the population have brown eyes. If 15 people are randomly​ selected, find the probability that at least 13 of them have brown eyes. Is it unusual to randomly select 15 people and find that at least 13 of them have brown​ eyes? Why or why​ not? The probability that at least 13 of the 15 people selected have brown eyes is nothing. ​(Round to three decimal places as​ needed.) Is it unusual to randomly select 15 people and find that at least 13 of them have brown​ eyes? Note that a small probability is one that is less than 0.05. A. No​, because the probability of this occurring is very small. B. Yes​, because the probability of this occurring is not small. C. Yes​, because the probability of this occurring is very small. D. No​, because the probability of this occurring is not small.

Answer 1

a)
`P(X \geq 13) = P(X=13) +P(X=14) +P(X=15)`

`P(X=13)=(15C13)(0.9)^(13) (1-0.9)^(15-13)=0.267`

`P(X=14)=(15C14)(0.9)^(14) (1-0.9)^(15-14)=0.343`

`P(X=15)=(15C15)(0.9)^(15) (1-0.9)^(15-15)=0.206`

And replacing we got:

`P(X \geq 13) = P(X=13) +P(X=14) +P(X=15)=0.267+0.343+0.206=0.816`

b) D. No​, because the probability of this occurring is not small.

Explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Part a

If 15 people are randomly​ selected, find the probability that at least 13 of them have brown eyes.

Let X the random variable of interest, on this case we now that:

`X \sim Binom(n=15, p=0.90)`

The probability mass function for the Binomial distribution is given as:

`P(X)=(nCx)(p)^x (1-p)^(n-x)`

Where (nCx) means combinatory and it's given by this formula:

`nCx=(n!)/((n-x)! x!)`

And we want to find this probability:

`P(X \geq 13) = P(X=13) +P(X=14) +P(X=15)`

`P(X=13)=(15C13)(0.9)^(13) (1-0.9)^(15-13)=0.267`

`P(X=14)=(15C14)(0.9)^(14) (1-0.9)^(15-14)=0.343`

`P(X=15)=(15C15)(0.9)^(15) (1-0.9)^(15-15)=0.206`

And replacing we got:

`P(X \geq 13) = P(X=13) +P(X=14) +P(X=15)=0.267+0.343+0.206=0.816`

Part b

Is it unusual to randomly select 15 people and find that at least 13 of them have brown​ eyes? Note that a small probability is one that is less than 0.05.

Since our calculated probability is too higher compared to 0.05 we can conclude this:

D. No​, because the probability of this occurring is not small.