Question

5. Waste Management, Inc., (WMI) claims that it collects curbside trash from 80% of its customers before 9 a.m. A community watchdog group surveys 100 randomly selected customers of WMI, and asks whether their trash is collected before 9 a.m. The number of customers who reply "Yes" is a binomial random variable. a. Find and illustrate the probability that 80 customers reply "Yes." b. Find and illustrate the probability that fewer than 75 reply "Yes."

Answer 1

Answer:-

Let random variable, X = The customer selected in watch group by WMI.

Thus,

$X \sim Binomial (n,p)$

Here, $n=100$ &

p= 1/2

Here is P is probability of success in a single Bernoulli trial. Note that X is Bernoulli random variable since it only two outcomes either "YES" or "NO" whether their tash collected before 9.a.m. or not.

Hence,       

p= 1/2

We kno pdf of $X \sim Binomial (n,p)$ is given by,

$p(x)=\binom{n}{x}*p^x*{(1-P)^{(n-x)}}$   

ST= 1,2, 3, ....... 100

Where,

  $\binom{n}{x}=\frac{n!}{x!*(n-x)!}$

Thus,

a).

To find the probability that 80 customers reply "YES".

  $i.e. P(x=80)=\binom{100}{80}*p^{(80)}*{(1-P)^{(100-80)}}$

Here, p=1/2

$i.e. P(x=80)=\binom{100}{80}*(1/2)^{(80)}*{(1-1/2)^{(20)}}$

  $i.e. P(x=80)=\frac{100!}{80!*(100-80)!}*(1/2)^{(80)}*{(1/2)^{(20)}}$

$i.e. P(x=80)=\frac{100!}{80!*20!}*(1/2)^{(80)}*{(1/2)^{(20)}}$

  

This imply that chance that 80 customers reply yes is nearly 0.000000042%

b).

To find the probability that fewer that 75 customers reply "YES".

  $i.e. P(x<75)=\sum_{x=1}^{74}\frac{100!}{x!*(100-x)}*(1/2)^{(x)}*{(1/2)^{(100-x)}}$

Note that this is Cumulative Distribution Function (CDF) of X at x=74.

Thus,

  

This imply that chance that less than 75 customers reply yes is nearly 99.99%