Question Type 1: Finding the conditional probability Exercises

Let $B_1,B_2,B_3$ be disjoint with $P(B_1)=0.5$, $P(B_2)=0.2$, $P(B_3)=0.3$. If $P(A\mid B_1)=0.6$, $P(A\mid B_2)=0.3$, $P(A\mid B_3)=0.1$, determine $P(B_1\mid A)$.

Three mutually exclusive events $B_1,B_2,B_3$ satisfy $P(B_1)=0.2$, $P(B_2)=0.5$, $P(B_3)=0.3$. Given $P(A\mid B_1)=0.2$, $P(A\mid B_2)=0.5$, $P(A\mid B_3)=0.8$, find $P(B_3\mid A)$.

Servers A, B and C handle 25%, 50% and 25% of network traffic respectively. Their failure rates are 2%, 5% and 4%. If a request fails, what is the probability it was handled by server B?

Given $P(A\cap B)=0.12$, $P(B)=0.3$ and $P(A)=0.4$, find $P(B\mid A)$.

Box 1 contains 3 red and 2 blue balls. Box 2 contains 1 red and 4 blue balls. A box is chosen at random with $P(\text{Box 1})=0.6$ and $P(\text{Box 2})=0.4$. If the drawn ball is red, what is the probability it came from Box 1?

Given $P(A\mid B_1)=0.7$, $P(A\mid B_2)=0.2$, $P(B_1)=0.3$ and $P(B_2)=0.7$, find $P(B_1\mid A)$.

If $P(A)=0.2$, $P(B)=0.5$ and $P(A\cap B)=0.1$, calculate $P(A\mid B)$.

In a town, 30% of households own cats, 20% own dogs and 10% own both. If a household is known to own at least one of these pets, what is the probability it owns a cat?

A medical test for a disease is 95% sensitive and has a 10% false positive rate. If the disease prevalence is 5%, what is the probability that a person actually has the disease given a positive test result?

If $P(A\mid B_1)=0.4$, $P(A\mid B_2)=0.6$, and $P(B_1)=P(B_2)=0.5$, calculate $P(B_2\mid A)$.

Suppose $P(A\mid B)=0.7$, $P(A\mid B^c)=0.2$, and $P(B)=0.4$.

Find $P(B\mid A^c)$.

A drug screening test is 90% sensitive and has a 5% false positive rate. If 2% of the population uses the drug, find the probability that a randomly tested individual is actually a user given a positive result.