Updating beliefs from evidence. The results to know for quant trading interviews, then a worked example of each of the 5 question types. Free, no sign-up.
Practise endless questions like these
Try each one before opening the solution. In the app, every type generates fresh numbers each time.
Base rates (medical test)
A condition affects 5% of people. A test detects it 95% of the time when present and gives a false positive 2% of the time when absent. Someone tests positive. What is the probability they have the condition?
Answer: 5/7 ≈ 0.71429
Odds-form Bayes
A bag has 10 coins: 9 fair and 1 double-headed. You pick one at random and flip it 6 times, getting 6 heads. What is the probability you picked the double-headed coin?
Answer: 0.87671
Conditioning on "at least one"
A family has 4 children. Given at least one is a boy, what is the probability all 4 are boys? (Each child is independently a boy or girl with probability 1/2.)
Answer: 1/15 ≈ 0.066667
Conditioning on dice sums
Two dice are rolled and you are told the sum is at least 7. What is the probability at least one die shows a six?
Answer: 11/21 ≈ 0.52381
Two-urn Bayes
Urn A has 3 red and 4 blue balls; urn B has 6 red and 5 blue. You pick an urn with a fair coin and draw one ball: it's red. What is the probability it came from urn A?
Answer: 0.44
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