CTJan27 Online - JMSS Exam Prep Set Theory Two Way Tables and Tree Diagrams
Multiple Choice
A survey of 100 students showed that 60 play Soccer ($S$) and 35 play Netball ($N$). 20 students play both. If this data is organized into a two-way table structure, how many students would be categorized as playing Soccer only ($S \cap N'$)?
A population of 100 people is analyzed. A tree diagram shows that 40 people are Male ($M$) and 60 are Female ($F$). If 10 people are categorized as Male and having blonde hair ($B$), how many people are Male but do NOT have blonde hair ($M \cap B'$)?
In a class of 30 students, 18 like Apples ($A$) and 15 like Bananas ($B$). 5 students like neither fruit. How many students like both Apples and Bananas ($A \cap B$)?
A two-way table tracks students who own a Phone ($P$) or a Laptop ($L$). The four sections contain the following counts: $n(P \cap L) = 45$, $n(P' \cap L) = 10$, $n(P \cap L') = 25$, and $n(P' \cap L') = 5$. What is the total number of students surveyed $n(\xi)$?
Let $\xi = \{1, 2, 3, ..., 10\}$. Let $E$ be the set of even numbers, and $P$ be the set of prime numbers. If a tree diagram is constructed based on these sets, how many elements belong to the final branch representing odd prime numbers ($E' \cap P$)?
Given a universal set $\xi$ of 150 items. Set $X$ contains 90 items. What is the total number of items in the complement of $X$, denoted $X'$?
A two-way table summarizes 120 cars based on color (Red ($R$) or Not Red ($R'$)) and size (Large ($L$) or Small ($L'$)). The entry found at the intersection of the row $R$ and the column $L$ is 35. What set relationship does the value 35 represent?
Out of 50 students, 30 are enrolled in Art ($A$), 25 are enrolled in Music ($M$), and 10 are enrolled in both. If this data is displayed in a tree structure, what is the count of students categorized under the final branch representing neither subject ($A' \cap M'$)?
A tree diagram starts with 100 customers. The trunk splits into those who bought item $A$ (70 customers) and those who did not ($A'$, 30 customers). The $A$ branch then splits, showing 40 customers bought item $B$ (i.e., $A \cap B$). Which numerical value should be placed on the branch representing customers who bought $A$ but not $B$ ($A \cap B'$)?
Let $\xi$ be the set of letters of the English alphabet. Let $L$ be the set of unique letters in the word 'TRIGONOMETRY'. If $L$ is mapped onto a set structure, what is the total number of elements $n(L)$?
A two-way table tracks two sets, $A$ and $B$. We are given $n(A)=50$, $n(B)=40$, and $n(A \cap B)=15$. What value represents the region $A \cap B'$ (A only)?
In constructing a set tree diagram, the initial trunk splits the universal set $\xi$ into two main branches, $X$ and $X'$. If $n(\xi)=200$ and $n(X')=120$, what value must be written on the $X$ branch?
In a group of 80 tourists, 50 speak Spanish ($S$), 40 speak French ($F$), and 15 speak both. If this information is used to map sets, how many tourists speak Spanish OR do NOT speak French ($S \cup F'$)?
Using the structure of a two-way table (or Karnaugh map structure) for sets $A$ and $B$, the cells contain: $n(A \cap B)=10$, $n(A \cap B')=20$, $n(A' \cap B)=5$, and $n(A' \cap B')=15$. What is the total count of elements belonging to Set $B$?
Consider $\xi$ as the set of letters in the word 'GEOGRAPHY'. $V$ is the set of vowels, and $C$ is the set of consonants. How many letters in 'GEOGRAPHY' belong to the intersection $V \cap C'$?
A tree diagram shows that 120 people bought a book ($B$) and 80 people did not ($B'$). Of those who bought a book, 30 also bought a magazine ($M$). If the total surveyed was 200, what percentage of the total population bought both the book AND the magazine?
In a universal set where $n(\xi)=90$, 70 own a bicycle ($B$) and 50 own rollerblades ($R$). If every person owns at least one of the items ($n(B \cup R) = 90$), how many people own both items ($B \cap R$)?
A two-way table is constructed based on two criteria: $H$ (Has a pet) and $G$ (Wears glasses). The cell corresponding to $H'$ and $G$ contains the number 25. Which statement accurately describes the 25 people?
A survey analyzed $150$ students. $80$ students play Soccer ($S$) and $65$ play Basketball ($B$). If $20$ students play neither sport, what is the value corresponding to the intersection $S \cap B$ in a completed two-way table for this data?
A scientist classifies $500$ insects using a two-stage tree diagram based on Species (A or B) and Gender (Male $M$ or Female $F$). If $60\\%$ belong to Species A, and $30\\%$ of Species A are Female, while $70\\%$ of Species B are Male, how many insects are Male in total?
In a class of $100$ students, $40$ study Art ($A$), $70$ study Biology ($B$), and $15$ study neither subject. If this data is organized into a two-way table, what is the count for students who study Art but not Biology (i.e., $A \cap B^c$)?
A two-stage tree diagram analyzes $200$ computer students based on Pass ($P$) or Fail ($P^c$), and then by course level (Advanced ($A$) or Standard ($S$)). Given that $120$ students passed, $40$ students failed the test and took the Standard course ($P^c \cap S$), and $100$ students total took the Advanced course ($A$), how many students passed the test and took the Standard course ($P \cap S$)?
In a survey of $100$ students regarding three options $X, Y,$ and $Z$, $10$ students selected none. The raw intersections were: $|X \cap Y| = 25$, $|Y \cap Z| = 20$, and $|X \cap Z| = 15$. If $5$ students selected all three options ($X \cap Y \cap Z$), what number represents students who selected *exactly two* options?
A two-stage tree diagram classifies $300$ objects. The first stage separates $180$ Red objects ($R$) from $120$ Blue objects ($B$). The second stage classifies them by Shape (Square $S$ or Circle $C$). If $70\\%$ of the Red objects are Square, and $20\\%$ of the Blue objects are Circular, what is the total number of Circular objects?
A two-way table classifies $400$ employees by Location (North $N$ or South $S$) and Transport (Car $C$ or Bus $B$). The following cells are known: $150$ employees work North and use a Car ($N \cap C$), and $120$ employees work South and use a Bus ($S \cap B$). If $200$ employees total use the Bus, what is the total number of employees who work in the North ($|N|$)?
A Grade 9 cohort of $200$ students is analyzed using a tree diagram. The first branch is Science ($S$) (80 students) or Arts ($A$). The second branch indicates passing ($P$) a compulsory test. Given that $70\\%$ of Science students passed, and $25\\%$ of Arts students failed the test, determine the total count of students who failed the compulsory test ($P^c$).
A population of $300$ items is classified using criteria $X$ and $Y$. The number of items meeting criterion $X$ is twice the number of items not meeting $X$. The number of items not meeting $Y$ ($|Y^c|$) is $50$ less than the number of items meeting $Y$ ($|Y|$). If $40$ items meet neither $X$ nor $Y$ ($|X^c \cap Y^c| = 40$), what is the number of items that meet both criteria ($|X \cap Y|$)?
A completed two-stage tree diagram shows the following final outcomes for a population: $|A \cap B| = 60$, $|A \cap B^c| = 30$, $|A^c \cap B| = 45$, and $|A^c \cap B^c| = 15$. If the total population is used as the root node, what percentage of the total population satisfies criterion $B$?