Question

an employee at the department of public transportation is tasked with determining bus - usage patterns among city residents according to neighborhood and type of pass purchased. below are the results of her study of a sampling of bus users. assume that each bus user purchases only one type of pass. rows represent the neighborhoods where bus riders live. columns represent the types of passes purchased by bus riders.
what is the probability of purchasing a monthly pass or living in forest lake?
use the frequency table to compute the following.
probability of purchasing a monthly pass: p(a)=
probability of living in forest lake: p(b)=
probability of purchasing a monthly pass and living in forest lake: p(a and b)=
probability of purchasing a monthly pass or living in forest lake: p(a or b)=
event a: purchasing a monthly pass
event b: living in forest lake

	forest lake	ducktown	rudolph	rio vista	total
ten ride	42	46	12	8	108
monthly	48	41	33	45	167
total	110	105	54	61	330

Explanation:

Step1: Recall probability formula

The probability of an event $E$ is given by $P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$.

Step2: Calculate $P(A)$

Event $A$ is purchasing a monthly - pass. The total number of monthly - pass purchases is $167$. The total number of all pass - purchases is $330$. So $P(A)=\frac{167}{330}$.

Step3: Calculate $P(B)$

Event $B$ is living in Forest Lake. The total number of people living in Forest Lake is $110$. So $P(B)=\frac{110}{330}=\frac{1}{3}$.

Step4: Calculate $P(A\cap B)$

The number of people who both purchase a monthly pass and live in Forest Lake is $48$. So $P(A\cap B)=\frac{48}{330}=\frac{8}{55}$.

Step5: Calculate $P(A\cup B)$

Use the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Substitute the values: $P(A\cup B)=\frac{167}{330}+\frac{110}{330}-\frac{48}{330}=\frac{167 + 110-48}{330}=\frac{229}{330}$.

Answer:

$P(A)=\frac{167}{330}$
$P(B)=\frac{1}{3}$
$P(A\cap B)=\frac{8}{55}$
$P(A\cup B)=\frac{229}{330}$