Question Type 1: Constructing Venn diagrams to calculate probabilities for 2 events Exercises

A fair coin is flipped. If it lands heads, a biased coin with $P(H) = 0.75$ is tossed. If it lands tails, a different biased coin with $P(H) = 0.25$ is tossed. Find the probability of getting heads on the second toss.

Given events $A$ and $B$ with $P(A)=0.7$, $P(B)=0.4$, and $P(A \cap B)=0.3$, find $P(A \cup B)$.

In a class of $80$ pupils, $30$ passed Maths, $50$ passed English and $10$ passed both. Find the probability that a randomly selected pupil passed at least one of the two subjects.

A person chooses Yes or No with equal probability. If they choose Yes, the next choice is Yes with probability $0.75$; if they choose No, the next choice is Yes with probability $0.25$. Find the probability that they choose Yes then No.

In a community of 500 people, 200 own a car, 150 own a bicycle and 100 own both. Calculate the probability that a randomly chosen person owns neither.

A delivery follows one of two routes: Route A with probability 0.3 or Route B with probability 0.7. On each route, on-time delivery occurs with probability 0.8 for Route A and 0.6 for Route B. If the delivery is on time, the probability of customer satisfaction is 0.9; if it is late, the probability of customer satisfaction is 0.3.

Calculate the probability that a customer is satisfied.

Two fair six-sided dice are rolled. Find the probability that the product of the two numbers shown is $12$.

In a survey of 50 students, 20 study French, 15 study German, and 5 study both. Calculate the probability that a randomly chosen student studies neither language.

A red die and a blue die are rolled. Find the probability that the red die shows a higher number than the blue die.

A factory uses machine $A$ $60\%$ of the time (defect rate $5\%$) and machine $B$ $40\%$ of the time (defect rate $2\%$). Calculate the probability that a randomly chosen item is defective.