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

In a survey of 100 people, 60 liked tea, 45 liked coffee and 20 liked both. Using a Venn diagram interpretation, find the probability that a randomly chosen person likes coffee.

In a Venn diagram of four mutually overlapping sets it is found $P(\text{none of the events})=0.1$. If $P(\text{at least one})=0.9$, verify by interpretation of the diagram that these two values sum to 1.

In a survey, 40% of people exercise regularly, 25% follow a diet plan and 15% do both. What is the probability that a person exercises but does not follow the diet plan? Illustrate with a Venn diagram.

In a class of 50 students, 30 study Math, 20 study Science and 10 study both. Using a Venn diagram approach, what is the probability a student studies neither subject?

Given a Venn diagram showing two events $A$ and $B$ in a sample space $S$ with $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$, calculate $P(A\cup B)$.

Given two events $A$ and $B$ with $P(A)=0.5$, $P(B)=0.6$, and $P(A\cup B)=0.8$, use a Venn diagram to find $P(A\cap B)$.

A probability tree describes two independent stages: stage 1 succeeds with probability 0.6 and fails with probability 0.4; stage 2 succeeds with probability 0.7 and fails with probability 0.3. Calculate the probability that at least one stage succeeds.

Given a rectangular probability diagram partitioned into regions for events $X$ and $Y$ with joint probabilities $P(X\cap Y)=0.1$, $P(X\cap Y^c)=0.2$, $P(X^c\cap Y)=0.3$, find $P(X^c\cup Y)$.

A test has a pass probability of 0.7. If you fail you retake once, with the same pass probability. Draw the tree and find the probability of passing at least once.

A probability tree for a medical test shows disease prevalence 5%, test gives true positive with 90% and false positive with 8%. Find the probability a randomly tested person tests positive.

A tree diagram models three outcomes on first roll of a loaded die: 1–3 with prob. 0.5, 4–5 with prob. 0.3, 6 with prob. 0.2; if 1–3 occurs, second roll has success prob. 0.4 else success prob. 0.6. Find probability of success in second roll.

Three events $A,B,C$ in a Venn diagram have $P(A)=0.4$, $P(B)=0.5$, $P(C)=0.3$, $P(A\cap B)=0.1$, $P(A\cap C)=0.05$, $P(B\cap C)=0.07$, and $P(A\cap B\cap C)=0.02$. Compute $P(A\cup B\cup C)$.