Question Type 2: Constructing Venn diagrams to calculate probabilities for 3 or more events Exercises

If $P(A)=0.5$, $P(B)=0.6$, $P(C)=0.4$, $P(A\cap B)=0.3$, $P(A\cap C)=0.2$, $P(B\cap C)=0.25$ and $P(A\cup B\cup C)=0.9$, find $P(A\cap B\cap C)$.

A conference has 200 attendees. The sessions are morning ($M$), afternoon ($A$), and evening ($E$). The attendance is as follows:

• 60% attend $M$
• 50% attend $A$
• 40% attend $E$
• 30% attend both $M$ and $A$
• 25% attend $M$ and $E$
• 20% attend $A$ and $E$
• 10% attend all three sessions

How many attendees attend only the morning session?

If $P(A)=0.7$, $P(B)=0.5$, $P(C)=0.6$, $P(A\cap B)=0.4$, $P(A\cap C)=0.35$, $P(B\cap C)=0.3$ and $P(A\cap B\cap C)=0.2$, find the conditional probability $P(A\mid B\cup C)$.

Given events $A$, $B$ and $C$ satisfy $P(A)=0.6$, $P(B)=0.5$, $P(C)=0.4$, $P(A\cap B)=0.3$, $P(A\cap C)=0.2$, $P(B\cap C)=0.15$ and $P(A\cap B\cap C)=0.1$, calculate $P(A\cup B\cup C)$.

In a survey of device usage, $P(\text{phone})=0.7$, $P(\text{tablet})=0.5$, $P(\text{laptop})=0.6$, $P(\text{phone}\cap\text{tablet})=0.4$, $P(\text{phone}\cap\text{laptop})=0.45$, $P(\text{tablet}\cap\text{laptop})=0.35$, and $P(\text{all three})=0.25$. Calculate the probability that a randomly selected respondent uses at least two types of device.

Given $P(A)=0.5$, $P(B)=0.6$, $P(C)=0.55$, $P(A\cap B)=0.3$, $P(A\cap C)=0.25$, $P(B\cap C)=0.35$, and $P(A\cap B\cap C)=0.2$, calculate the probability that exactly one of the events occurs.

In a class, 80% pass English ($E$), 70% pass Maths ($M$), 60% pass Science ($S$), with $P(E\cap M)=0.65$, $P(E\cap S)=0.5$, $P(M\cap S)=0.45$ and $P(E\cap M\cap S)=0.4$. Calculate the percentage that fail all three subjects.

With $P(A)=0.6$, $P(B)=0.5$, $P(C)=0.4$, $P(A\cap B)=0.3$, $P(A\cap C)=0.2$, $P(B\cap C)=0.15$ and $P(A\cap B\cap C)=0.1$, calculate the probability that exactly one of the events occurs.

A single card is drawn at random from a standard 52-card deck. Let $A$ be the event the card is red, $B$ the event it is a face card, and $C$ the event it is a spade. Given $P(A)=\frac{1}{2}$, $P(B)=\frac{12}{52}$, $P(C)=\frac{1}{4}$, $P(A\cap B)=\frac{6}{52}$, $P(A\cap C)=0$, $P(B\cap C)=\frac{3}{52}$, and $P(A\cap B\cap C)=0$, find the probability that exactly one of the events $A,B,C$ occurs.

Using $P(A)=0.6$, $P(B)=0.5$, $P(C)=0.4$, $P(A \cap B)=0.3$, $P(A \cap C)=0.2$, $P(B \cap C)=0.15$ and $P(A \cap B \cap C)=0.1$, find the probability that exactly two of the events occur.

Three medical tests for a disease have probabilities of positive results $P(T_1)=0.15$, $P(T_2)=0.15$, $P(T_3)=0.20$, with $P(T_1\cap T_2)=0.05$, $P(T_1\cap T_3)=0.08$, $P(T_2\cap T_3)=0.06$, and $P(T_1\cap T_2\cap T_3)=0.02$. Calculate the probability that at least one test is positive.

A company sells products A, B and C. Market research gives $P(A)=0.5$, $P(B)=0.4$, $P(C)=0.3$, $P(A\cap B)=0.2$, $P(A\cap C)=0.15$, $P(B\cap C)=0.1$, $P(A\cap B\cap C)=0.05$. What is the probability that a random customer buys exactly two of these products?