In this topic you count outcomes and divide. When every outcome is equally likely, the probability of an event is the number of outcomes in the event divided by the total number of outcomes. The whole problem is two counts and one division. A problem belongs here when it says uniformly at random, well shuffled, or drawn at random without replacement. Those phrases mean every outcome is equally likely. You will practice: the multiplication rule, permutations, combinations, probability as a ratio of counts, inclusion-exclusion, and complements.
A quality-control lab receives 8 different smartphone prototypes, but its testing rig has only 3 slots: one for drop testing, one for battery testing, and one for water testing. Each slot must be filled with a different prototype. In how many ways can the inspector assign prototypes to the three slots?
Answer:
The three slots are distinct roles, so order matters. Count ordered selections of 3 from 8.
A genetics lab maintains 12 distinct fruit-fly lines and has funding to send exactly 5 of them for full genome sequencing. The 5 lines are sequenced together as one batch, so the order of selection does not matter. How many different batches of 5 lines can the lab choose?
Answer:
An unordered selection of 5 from 12 is a combination.
A swim coach has 10 swimmers and must assign 4 of them to the four legs of a medley relay. The order matters: each of the 4 chosen swimmers swims a specific leg (backstroke, breaststroke, butterfly, freestyle). How many different relay lineups are possible?
Answer:
The lineup is built in 4 ordered slots with the choices multiplying. This is the permutation of 4 from 10.
A botanist monitors 9 distinct orchid species and has resources to plant exactly 4 of them together in a single conservation plot. The plot is one unordered group, only which species are included matters. How many different sets of 4 species can she choose?
Answer:
The plot is an unordered selection of 4 species from 9, so the count is a combination.
A bistro offers a fixed-price dinner: diners pick one of 4 starters, one of 6 mains, and one of 3 desserts. How many different three-course dinners are possible?
Answer:
The three choices are made independently in sequence, so multiply by the multiplication rule.
An animal shelter has 9 volunteers, and exactly 2 of them must be chosen to staff the Saturday adoption booth. The two booth roles are identical, so only which pair is chosen matters. How many different pairs of volunteers are possible?
Answer:
Order does not matter, so count unordered pairs.
A souvenir stand sells 5 different T-shirt designs and 3 different cap colors. A tourist buys exactly one T-shirt and one cap. How many different T-shirt and cap combinations are possible?
Answer:
Multiply the number of independent choices.
A bag contains 3 red marbles and 7 blue marbles. One marble is drawn uniformly at random. What is the probability that it is red?
Answer:
Favorable outcomes over total outcomes.
A student council has 7 members. They must fill three different positions: president, vice president, and treasurer. No member can hold more than one position. In how many ways can the three positions be filled?
Answer:
Fill the positions one at a time without reusing a member.
A hiking club lists 8 possible trails for the season. It must choose 3 of them for the schedule. The order of the chosen trails does not matter. How many different choices are possible?
Answer:
Order does not matter, so count combinations.
A drawer holds 10 batteries. Exactly 4 of them are dead and the other 6 work. You grab 2 batteries uniformly at random without replacement. What is the probability that both batteries work?
Answer:
Count pairs of working batteries over all pairs.
In a town survey of 50 households, 30 subscribe to streaming service A, 22 subscribe to streaming service B, and 12 subscribe to both. One household is picked uniformly at random. What is the probability that it subscribes to at least one of the two services?
Answer:
Use inclusion-exclusion on the two subscriber sets, then divide by 50.
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