an employee at the department of public transportation is tasked with determining bus - usage patterns among…

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.\nwhat is the probability of purchasing a monthly pass or living in forest lake?\nuse the frequency table to compute the following.\nprobability of purchasing a monthly pass: p(a)=\nprobability of living in forest lake: p(b)=\nprobability of purchasing a monthly pass and living in forest lake: p(a and b)=\nprobability of purchasing a monthly pass or living in forest lake: p(a or b)=\nevent a: purchasing a monthly pass\nevent b: living in forest lake\n| | forest lake | ducktown | rudolph | rio vista | total |\n|--|--|--|--|--|--|\n| single ride | 20 | 18 | 9 | 8 | 55 |\n| ten ride | 42 | 46 | 12 | 8 | 108 |\n| monthly | 48 | 41 | 33 | 45 | 167 |\n| total | 110 | 105 | 54 | 61 | 330 |

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.\nwhat is the probability of purchasing a monthly pass or living in forest lake?\nuse the frequency table to compute the following.\nprobability of purchasing a monthly pass: p(a)=\nprobability of living in forest lake: p(b)=\nprobability of purchasing a monthly pass and living in forest lake: p(a and b)=\nprobability of purchasing a monthly pass or living in forest lake: p(a or b)=\nevent a: purchasing a monthly pass\nevent b: living in forest lake\n| | forest lake | ducktown | rudolph | rio vista | total |\n|--|--|--|--|--|--|\n| single ride | 20 | 18 | 9 | 8 | 55 |\n| ten ride | 42 | 46 | 12 | 8 | 108 |\n| monthly | 48 | 41 | 33 | 45 | 167 |\n| total | 110 | 105 | 54 | 61 | 330 |

Answer

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:

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