Question Type 7: Given values of events, using diagrams to interpret probabilities Exercises

In a survey of 150 students, 90 study mathematics. What is the probability that a randomly selected student studies mathematics?

Given three events $A$, $B$, $C$ with $P(A)=0.4$, $P(B)=0.5$, $P(C)=0.3$, $P(A\cap B)=0.2$, $P(A\cap C)=0.15$, $P(B\cap C)=0.1$, and $P(A\cap B\cap C)=0.05$, find the probability that exactly two of the events occur.

Two events A and B satisfy $P(A)=0.6$, $P(B)=0.5$, and $P(A\cap B)=0.3$. What is $P(A\Delta B)$, the probability that exactly one of A or B occurs?

Find the probability that a randomly selected student studies exactly one of the three subjects.

In a class of 100 students, 60 study maths ($M$), 45 study science ($S$), and 35 study history ($H$). There are 25 who study both maths and science, 15 who study both maths and history, 10 who study both science and history, and 5 who study all three.

What is the probability a randomly selected student studies at least one of these subjects?

In a group of 100 students, 70 play soccer, 40 play basketball, and 25 play both sports. What is the probability that a randomly chosen student plays neither sport?

In a survey of 500 employees, 200 smoke, 300 drink alcohol, and 100 both smoke and drink. What is the probability that an employee smokes or drinks but not both?

In a sample of 250 people, 160 own a car, 120 own a bike, and 70 own both. Given that a person owns at least one of these, what is the probability they own both?

In a survey of 150 students, 90 study mathematics, 70 study physics, and 30 study both subjects. What is the probability that a randomly selected student studies mathematics or physics?

In a survey of 150 people, 90 like chocolate, 60 like vanilla, and 30 like both. What is the conditional probability that a person likes chocolate given that they like vanilla?

In a class of 80 students, 50 take French, 30 take German, and 10 take both. What is the probability that a student takes German but not French?

In a survey of 150 students, 90 study mathematics, 70 study physics, and 30 study both subjects.

Find the probability that a randomly selected student studies neither mathematics nor physics.