You work with events and their probabilities. An event is a set of outcomes. Probabilities follow three rules. Every probability is at least 0. The whole sample space has probability 1. Disjoint events add. You use these rules to compute probabilities of combined events. A problem belongs here when it gives probabilities of named events and asks about "or", "not", "at least one", or "exactly one". No random variables appear. You will practice: the complement rule, inclusion-exclusion, cut into disjoint pieces.
At a campus coffee cart, the number of customers waiting in line at 9 a.m. is 0 with probability 0.22, exactly 1 with probability 0.31, and exactly 2 with probability 0.28. What is the probability that at least 3 customers are waiting?
Answer:
The events for 0, 1, and 2 customers waiting are disjoint, so their probabilities add. The answer then follows by the complement rule.
In a population of pea plants, a randomly chosen plant has purple flowers with probability 0.6, has wrinkled seeds with probability 0.5, and has both traits with probability 0.35. What is the probability that a randomly chosen plant has purple flowers or wrinkled seeds (or both)?
Answer:
Apply the addition rule for two events that can overlap.
The organizers of a three-day outdoor music festival are told by a forecaster that the probability it rains at least once during the festival is 0.85. What is the probability that the festival sees no rain at all?
Answer:
The event no rain at all is exactly the complement of rains at least once.
In a city marathon, a randomly chosen runner finishes in under 4 hours with probability 0.45, wears a running-club jersey with probability 0.30, and does both with probability 0.18. What is the probability that a randomly chosen runner finishes in under 4 hours or wears a club jersey (or both)?
Answer:
The two events can overlap, so use inclusion exclusion.
At a poultry farm, a randomly selected egg from the morning collection is cracked with probability 0.03. What is the probability that a randomly selected egg is not cracked?
Answer:
Not cracked is the complement of cracked, so apply the complement rule.
A ceramics studio glazes every mug it fires in exactly one of three colors: blue, green, or white. A randomly chosen mug is blue with probability 0.38 and green with probability 0.27. What is the probability that a randomly chosen mug is white?
Answer:
The three colors partition the sample space into disjoint pieces, so their probabilities sum to 1.
A florist plants a tulip bulb. The probability that it sprouts is 0.85. What is the probability that it does not sprout?
Answer:
Use the complement rule.
A carnival prize wheel stops on exactly one colored sector each spin. It stops on red with probability 0.2 and on gold with probability 0.5. What is the probability that it stops on red or gold?
Answer:
Red and gold are disjoint outcomes, so add their probabilities.
At a public library, a randomly chosen member borrowed a novel last month with probability 0.55, borrowed a biography with probability 0.35, and borrowed both kinds with probability 0.15. What is the probability that the member borrowed a novel or a biography last month?
Answer:
Apply inclusion-exclusion directly.
At a gym, a randomly chosen visitor uses the treadmills with probability 0.55 and uses the free weights with probability 0.45. The probability that the visitor uses at least one of the two is 0.8. What is the probability that the visitor uses both?
Answer:
Solve inclusion-exclusion for the intersection.
At a bakery, a randomly chosen loaf is sourdough with probability 0.4. The probability that it is sourdough and sells before noon is 0.28. What is the probability that it is sourdough and does not sell before noon?
Answer:
Split the sourdough event into two disjoint pieces by whether the loaf sells before noon, then subtract.
In an orchard, a randomly picked apple has a bruise with probability 0.3, has a worm hole with probability 0.2, and has both flaws with probability 0.1. What is the probability that the apple has neither flaw?
Answer:
Find the probability of at least one flaw with inclusion-exclusion, then take the complement.
Want these problems to schedule themselves? Inkerval brings handwritten practice and spaced repetition together on iPad.
Be first in when the doors open. No spam, ever.