Exercises: Two-way tables and independence

A high school surveyed $200$ students about whether they participate in after-school sports. The results are shown in the two-way frequency table below:

If one student is selected at random from this group, what is the probability that the student is a 10th grader who plays a sport? Express your answer as a decimal.

A company tracked the commuting methods and punctuality of its employees for a month to see if transportation type affects arrival times.

Find the conditional probability that an employee was on time, given that they took the bus.

The Venn diagram below shows the number of seniors at Washington High School who are in the Drama Club (Event $D$) and the Jazz Band (Event $J$).

If a senior is selected at random from this group, what is the probability that they are in the Drama Club OR the Jazz Band (or both)?

At a local diner in Chicago, the probability that a customer orders a burger is $0.65$, and the probability that a customer orders fries is $0.70$. The probability that a customer orders both a burger and fries is $0.55$. Use the addition rule to find the probability that a customer orders a burger or fries (or both).

A fair six-sided die is rolled, and a fair US quarter is flipped. Let $A$ be the event of rolling a number greater than $4$, and let $B$ be the event of the coin landing on heads. Assuming these events are completely independent, what is the exact probability of both $A$ and $B$ occurring?

A researcher is studying the relationship between gender and left-handedness in a sample of $300$ adults.

Calculate $P(\text{Left-Handed} \mid \text{Male})$ and $P(\text{Left-Handed})$. Based on comparing these probabilities, are the events "being male" and "being left-handed" independent?

A survey asked $500$ residents of a town about their primary source of daily news.

Let $I$ be the event that a randomly selected resident's primary news source is the Internet. Find the probability of the complement of $I$, denoted as $P(I^c)$.

A clinic uses a rapid test for a particular strain of flu. The two-way table shows the results of $1,000$ patients who were tested and later confirmed by a more accurate lab test.

A "false positive" occurs when a patient tests positive but does not actually have the flu. What is the probability of a false positive, given that the patient does not have the flu?

A city council is considering building a new public park. They surveyed residents to gauge support. Some of the data on the spreadsheet was accidentally deleted, leaving the incomplete table below:

First, determine the missing values $x$, $y$, and $z$. Then, if a surveyed resident is selected at random, find the probability that they are a renter, given that they support the park.

A university analyzed the choice of major among its $1,200$ incoming freshmen by gender to check for demographic trends.

Let $S$ be the event that a student chooses a STEM major, and $M$ be the event that the student is male. Calculate $P(S)$, $P(S \mid M)$, and $P(S \mid M^c)$. Are the events $S$ and $M$ independent? Explain your reasoning using the calculated probabilities.

The Venn diagram below represents a survey of $200$ high school students regarding their enrollment in Physics ($P$) and Chemistry ($C$).

Determine the probability that a randomly chosen student takes Physics, given that they are already enrolled in Chemistry. Express your answer as a fraction in simplest form.

In a standard deck of $52$ playing cards, let event $A$ be drawing a Heart and event $B$ be drawing a Face card (Jack, Queen, or King). The probability of drawing a Heart is $\frac{13}{52}$, and the probability of drawing a Face card is $\frac{12}{52}$. The probability of drawing a card that is a Heart OR a Face card is $\frac{22}{52}$.

Use the addition rule to find the probability of drawing a card that is BOTH a Heart and a Face card. Based on your result, are events $A$ and $B$ mutually exclusive?

The tree diagram below shows the probabilities of weather conditions in Seattle on a given autumn day, and whether an office worker, Alex, remembers to bring an umbrella.

Use the multiplication rule to find the probability that it rains AND Alex brings an umbrella. Then, find the total probability that Alex brings an umbrella, regardless of the weather.

A marketing firm created a two-way relative frequency table showing the relationship between age group and preferred streaming service for a sample of US adults.

If an adult from the survey is randomly selected and it is known they prefer Service B, what is the probability that they are over 35 years old? Round to the nearest thousandth.

At a car dealership, $60\%$ of the vehicles are SUVs, and $40\%$ are sedans. Of the SUVs, $30\%$ have four-wheel drive (4WD). Of the sedans, $10\%$ have 4WD. Construct a two-way probability table (or a tree diagram) to represent this situation. What is the probability that a randomly selected vehicle from this dealership has 4WD?

Let $A$ and $B$ be two events in a sample space. You are given that $P(A) = 0.55$, $P(B) = 0.40$, and $P(A \cap B) = 0.22$.

First, calculate $P(A \cup B)$. Then, determine if events $A$ and $B$ are independent, showing the specific mathematical condition you used to test for independence.

A tech company in Silicon Valley has $800$ employees. Among them, $500$ are engineers and $300$ are in sales/marketing. The HR department finds that $70\%$ of the engineers have a master's degree, while only $40\%$ of the sales/marketing employees have a master's degree.

Let $E$ be the event of selecting an engineer, and $M$ be the event of selecting an employee with a master's degree. Construct a two-way frequency table for this population. Then, mathematically prove whether events $E$ and $M$ are independent or dependent.

A rare disease affects $1$ in every $1,000$ people in a certain US state. A diagnostic test has been developed that is $98\%$ accurate for people who have the disease (sensitivity) and $95\%$ accurate for people who do not have the disease (specificity). The tree diagram below illustrates the probabilities.

If a person is chosen at random from the population, administered the test, and receives a positive result, what is the probability that they actually have the disease? Round your answer to the nearest tenth of a percent.

For two events $X$ and $Y$ in a sample space, it is known that $P(X) = 0.4$ and $P(X \cup Y) = 0.76$. If events $X$ and $Y$ are independent, find the exact value of $P(Y)$.

A quality control inspector at a factory is analyzing the defect rate of two different assembly lines. The results of yesterday's production are shown in the table below, where $k$ represents an unknown number of defective items from Line B.

Find the value of $k$ that would make the event "item is defective" perfectly independent of the event "item was produced on Line B".

The Venn diagram below displays the distribution of students who play video games ($V$) and read comic books ($C$). The number of students in each region is given as an algebraic expression in terms of $x$.

Given that the probability of selecting a student who plays video games is $P(V) = 0.40$, first solve for $x$. Then, find the conditional probability $P(C \mid V^c)$, which is the probability that a student reads comic books given that they do not play video games.

A bag contains $5$ red marbles, $4$ blue marbles, and $3$ green marbles. Two marbles are drawn at random without replacement. Let $R_1$ be the event that the first marble is red, and $R_2$ be the event that the second marble is red.

Calculate $P(R_1)$, $P(R_2 \mid R_1)$, and $P(R_1 \cap R_2)$. Then, calculate the total probability that the second marble is red, $P(R_2)$, by considering both cases for the first marble. Are $R_1$ and $R_2$ independent events?

A law firm is investigating a potential age discrimination case in a large corporation's hiring practices. They obtain the following data regarding applicants for management positions over the last two years:

Let $H$ be the event that an applicant is hired, and $O$ be the event that an applicant is $40$ years or older. Calculate $P(H)$, $P(H \mid O)$, and $P(H \mid O^c)$. In a legal context, if hiring decisions were completely independent of age, what exact number of applicants $40$ years or older would you have expected to be hired out of the $100$ who applied?

Two events $A$ and $B$ are such that $P(A) = 3 \cdot P(B)$. It is also given that $P(A \cap B) = 0.12$. If events $A$ and $B$ are independent, determine the exact values of $P(A)$ and $P(B)$. Then, calculate $P(A \cup B)$.