Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
The last problems compare business choices and find expected profit. Always subtract the cost when finding a net gain.
Standards: S-MD.A.3, S-MD.A.4, S-MD.B.5 · Level: Grade 12 Precalculus
Problem 1 Valid Distributions ★★★
Decide whether each table is a valid probability distribution. Explain.
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| (a) P(x) | 0.2 | 0.3 | 0.4 | 0.2 |
| (b) P(x) | 0.25 | 0.25 | 0.30 | 0.20 |
Problem 2 Missing Probability ★★★
A distribution has P(0) = 0.15, P(1) = 0.35, P(3) = 0.20, and an unknown P(2). Find P(2).
Problem 3 A Labeled Die ★★★
A fair cube has its six faces marked 0, 0, 1, 1, 1 and 4. Let X be the number on the top face. Write the distribution and find E(X).
Problem 4 Expected Value from a Table ★★★
Find E(X).
| x | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| P(x) | 0.4 | 0.3 | 0.2 | 0.1 |
Problem 5 Library Survey ★★★
Thirty students reported how many library books they borrowed last month: 0 books (6 students), 1 book (12), 2 books (9), 3 books (3). Build the empirical distribution and find the expected number of books.
Problem 6 Is It Fair? ★★★
A player pays $3 to play a game. With probability 14 the player wins $12, and otherwise wins nothing. Is the game fair?
Problem 7 School Raffle ★★★
A raffle sells 200 tickets at $5 each. The prizes are one $300 prize and two $50 prizes. Find the expected winnings and the expected net gain for one ticket.
Problem 8 Ten-Sector Spinner ★★★
The spinner costs $3 per spin and pays the amount in the sector it lands on. Write the distribution of the payout and find the expected net gain per spin.
Problem 9 Free Throws ★★★
A basketball player makes 70% of free throws, and two shots are independent. Let X be the number made. Find the distribution of X and E(X).
Problem 10 Insurance Policy ★★★
A policy costs $120 per year. There is a 3% chance of a $2,500 loss that the policy would cover. Find the expected loss without insurance and the expected net result of buying the policy.
Problem 11 Speaker Warranty ★★★
A $55 extended warranty on a speaker covers a repair that costs $180. The chance the speaker needs that repair is 12%. Compare the warranty price with the expected repair cost.
Problem 12 Comparing Two Games ★★★
Both games cost $2. Game A pays $5 with probability 0.3 and nothing otherwise. Game B pays $3 with probability 0.6 and nothing otherwise. Find the expected net gain of each and say which is better.
Problem 13 Televisions per Household ★★★
The chart shows the relative frequencies from a survey of 60 households. Find the expected number of televisions per household.
Problem 14 Two Four-Sided Dice ★★★
Two fair four-sided dice (numbered 1 to 4) are rolled, and X is the sum. Write the distribution and find E(X) and P(X ≥ 6).
Problem 15 Food Truck Location ★★★
A food truck owner expects sunny weather with probability 0.6 and rain otherwise. At Park A (open air) the truck earns $900 on a sunny day and $300 on a rainy day. At Plaza B (covered) it earns $600 on a sunny day and $550 on a rainy day. Find the expected earnings at each site and recommend one.
Problem 16 Insurer’s Profit ★★★
An insurer sells 5,000 policies at $90 each. Each policy has a 2% chance of a $3,000 claim. Find the expected claim cost per policy, the expected profit per policy and the expected total profit.
Problem 17 Guessing on a Quiz ★★★
A five-choice multiple-choice question scores +4 for a correct answer and −1 for a wrong answer. (a) Find the expected score when guessing at random. (b) After eliminating two choices, find the expected score when guessing among the other three. Should a student guess in each case?
Problem 18 Screening Parts ★★★
A factory ships parts that are defective 4% of the time. Each defective part shipped costs $25 in refunds. Screening each part costs $0.50 and finds every defective part. Compare the expected cost per part with and without screening and recommend an option.
Problem 19 Coin Flip Game ★★★
A game costs $4. The player flips three fair coins and wins $2 for each head. Write the distribution of the winnings and find the expected net gain.
Problem 20 Fundraiser Drawing ★★★
A fundraiser sells 400 tickets at $10 each. The prizes are one $1,000 prize, four $150 prizes and twenty $25 prizes. (a) Find the expected winnings and expected net gain for one ticket. (b) Find the fundraiser’s profit after paying all prizes.
Mis à jour le