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

In a sample of 500 outcomes, a Venn diagram for events $A, B, C$ has counts: only $A = 150$, only $B = 120$, only $C = 100$, $A \cap B$ only $= 50$, $A \cap C$ only $= 30$, $B \cap C$ only $= 20$, and all three $= 10$. Verify consistency of these counts and find the probability that at least two events occur.

Three events $A, B$, and $C$ are independent such that $P(A) = 0.4$, $P(B) = 0.5$, and $P(C) = 0.6$.

Calculate the probability that exactly one of these events occurs.

Given three events $A$, $B$, and $C$ such that $P(A)=0.6$, $P(B)=0.5$, $P(C)=0.4$, $P(A\cap B)=0.3$, $P(A\cap C)=0.2$, $P(B\cap C)=0.1$, and $P(A\cap B\cap C)=0.05$, calculate $P(A\cup B\cup C)$.

In a universal set of $150$ elements, a Venn diagram for events $A$, $B$, and $C$ has the following regions: only $A = 40$, only $B = 30$, only $C = 20$, $A ∩ B$ only $= 10$, $B ∩ C$ only $= 5$, all three $= 5$, and $A ∩ C$ only $= x$.

If every element lies in at least one of $A$, $B$, or $C$, find the value of $x$.

In a survey of 120 people, 70 like tea, 50 like coffee, and 30 like juice. If 20 like both tea and coffee, 15 like both tea and juice, 10 like both coffee and juice, and 5 like all three, how many people like at least one of the beverages?

In a class of $120$ students, $80$ study mathematics ($M$), $70$ study physics ($P$), and $60$ study chemistry ($C$). There are $50$ who study both $M$ and $P$, $40$ both $M$ and $C$, $30$ both $P$ and $C$, and $20$ students study all three subjects.

Find the number of students who study exactly two subjects.

In a universal set of 200 elements, subsets $A, B$, and $C$ satisfy $n(A) = 100$, $n(B) = 120$, $n(C) = 80$, $n(A \cap B) = 60$, $n(A \cap C) = 50$, $n(B \cap C) = 40$, and $n(A \cap B \cap C) = 30$.

Find the number of elements that lie in none of the subsets.

In a survey, respondents may choose any of three features $A$, $B$, and $C$. The following data represents the number of respondents in each region of a Venn diagram:

• $A$ only: $25$
• $B$ only: $20$
• $C$ only: $15$
• $A \cap B$ only: $10$
• $A \cap C$ only: $8$
• $B \cap C$ only: $5$
• All three features ($A \cap B \cap C$): $2$
• None of the features: $40$

Find the total number of respondents.

Given $P(A)=0.6$, $P(B)=0.5$, $P(C)=0.4$, $P(A\cap B)=0.3$, $P(A\cap C)=0.2$, $P(B\cap C)=0.15$, and $P(A\cap B\cap C)=0.1$, find the probability that none of the events occurs.

Events $A, B$ and $C$ satisfy the following conditions:

• $P(A) = 0.4$
• $P(B) = 0.5$
• $P(C) = 0.6$
• $P(A \cup B) = 0.7$
• $P(B \cup C) = 0.8$
• $P(A \cup C) = 0.75$
• $P(A \cup B \cup C) = 0.9$

Find $P(A \cap B \cap C)$.*

Given $P(A) = 0.6$, $P(B) = 0.7$, $P(C) = 0.5$, $P(A \cap B) = 0.45$, $P(A \cap C) = 0.35$, and $P(B \cap C) = 0.4$, determine the range of possible values for $P(A \cap B \cap C)$.