1. ## Statistics Probabilities Problems

1.) A survey of 500 college students revealed that 150 preferred morning classes, 250 preferred afternoon classes and 100 evening classes. Of those that preferred morning classes, 80 were business majors. Of those that preferred afternoon classes 150 were business majors, and of those that preferred evening classes, 20 were business majors.

c. What is the probability a student is a business major?
d. What is the probability a student prefers morning classes given that we know he/she is a business major?

With this information I was able to come up with the following tables and solutions:

I am pretty sure about a and b. Not so much with c and d though. Can anyone confirm if I'm doing these correctly?
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2.) Suppose that 15% of college students withdraw from school within their 1st yr of school.

a. Consider a group of 10 1st yr students. What's the probability that none of them withdraw from school within their 1st yr?

b. Of the same 10 students, what is the probability that all 10 withdraw sometime in their 1st yr?

c. What's the probability that fewer than 4 students withdraw?

d. How many of these 1st yr students are expected to withdraw in the 1st yr of school?
Use Expected value right?
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3.) Suppose that customers arrive in a restaurant at a rate of 10 customers per 30 minutes. [a poisson distribution problem]

a. What's the probability that 2 customers arrive in a 30-min period? [p(x=2)]
b. What's the probability that 6 or more customers arrive in a 30 min period? [p(x>=6) = 1-p(x<=5)]
c. What's the probability that 15 customers arrive in an hour?
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Thanks a bunch!

2. c. What is the probability a student is a business major?
$
\frac{250}{500}=0.5
$

d. What is the probability a student prefers morning classes given that we know he/she is a business major?
$
\frac{80}{250}=0.32
$

a. Consider a group of 10 1st yr students. What's the probability that none of them withdraw from school within their 1st yr?
$
(1-0.15)^{10}=0.197
$

b. Of the same 10 students, what is the probability that all 10 withdraw sometime in their 1st yr?
$
(0.15)^{10}=0.000
$
effectively
c. What's the probability that fewer than 4 students withdraw?
$
X\sim\mathrm B(10,0.15),\quad \mathrm P(X<4)=\mathrm P(X\leq 3)=0.9500
$
from binomial tables
d. How many of these 1st yr students are expected to withdraw in the 1st yr of school?
$
\mathrm E(X)=10\times0.15=1.5
$

3.) Suppose that customers arrive in a restaurant at a rate of 10 customers per 30 minutes. [a poisson distribution problem]

a. What's the probability that 2 customers arrive in a 30-min period? [p(x=2)]
b. What's the probability that 6 or more customers arrive in a 30 min period? [p(x>=6) = 1-p(x<=5)]
c. What's the probability that 15 customers arrive in an hour?
a. $X\sim\mathrm{Po}(10),\quad \mathrm P(X=2)=\mathrm e^{-10}\frac{10^2}{2!}=0.00227$

b. $\mathrm P(X\geq 6)=1-\mathrm P(X\leq 5)=0.93291$ from tables

c. $X\sim\mathrm{Po}(20),\quad \mathrm P(X=15)=\mathrm e^{-20}\frac{20^{15}}{15!}=0.05165$