Conditional probability is the probability that one event happens given that another event has already happened. It narrows the sample space to the cases where the given condition is true.
Once you know event B happened, you no longer look at every outcome. You only look at the part of the sample space where B is true, then ask how much of that smaller space also satisfies A.
Showing a random 20 of 80 problems.
Example 1
hard
Survey data: 25% of teens use Platform A only, 35% use Platform B only, 30% use both, 10% use neither. Find P(uses A∣uses B).
Example 2
medium
A test detects a disease in 95% of sick people and in 10% of healthy people. If 4% of the population has the disease, find P(positive test).
Example 3
challenge
In a random sample, 1% of subjects have a rare allele. A genetic test has 99% sensitivity and 98% specificity. If a randomly sampled subject tests positive, find P(allele∣positive).
Example 4
medium
A survey of 400 commuters found 250 take the bus, 150 take the train, and 80 take both. Find P(train∣bus).400 commuters: 250 take bus, 150 take train, 80 take both
Example 5
hard
A factory's data show P(defect∣Machine A)=0.02, P(defect∣Machine B)=0.05. Machines A and B produce 70% and 30% of items. Find P(Machine A∣defect).
Example 6
easy
A die shows a number greater than 3. What is P(it is a 5∣>3)?
Example 7
hard
Three coins are flipped. Given at least two heads, find P(all heads).
Example 8
medium
In a school, 60% of students play sports and 25% play sports AND music. What is P(music∣sports)?
Example 9
medium
Two dice are rolled. Given the sum is even, find P(sum=8).
Example 10
medium
Quality-control data: 5% of items are defective, 2% are both defective and from Shift 1, and 40% are from Shift 1. Find P(defective∣Shift 1).
Example 11
hard
In a population, P(college)=0.40, P(employed)=0.85, P(college and employed)=0.36. Find P(college∣employed) and compare with P(college).
Example 12
easy
In a class, 12 students play sports and 4 of those also play music. What is P(music∣sports)?12 students play sports; 4 of those also play music
Example 13
easy
A die is rolled. Given the result is less than 5, what is P(it is a 1)?
Example 14
medium
A sample of 300 customers: 120 own a smartphone, 60 own a tablet, 30 own both. Find P(smartphone∣tablet).300 customers: 120 own smartphone, 60 own tablet, 30 own both
Example 15
medium
In a random sample of 600 voters, 360 favor Candidate X. Among the 250 women sampled, 175 favor X. Find P(favors X∣woman).
Example 16
medium
In a class, 40% are seniors and 60% of seniors take calculus. What is P(senior and takes calculus)?
Example 17
medium
A bag has 4 white and 1 black. Two drawn without replacement. Given the first is white, what is P(second is black)?Bag: 4 white, 1 black — draw two without replacement
Example 18
hard
Bag A has 3 red and 7 white. Bag B has 6 red and 4 white. A bag is chosen at random and a ball drawn. Given the ball is red, find P(bag A).