Probability as Expectation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Probability as Expectation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Probability can be interpreted as the long-run relative frequency of an event over infinitely many identical trials of a random experiment.

P(heads)=0.5P(\text{heads}) = 0.5 means if you flip many times, about half will be heads.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: P(event)P(\text{event}) is the fraction of trials the event happens over very many repetitions.

Common stuck point: The procedure for probability as expectation is the easy part; the trap is expecting the exact expected count. Asking "Am I predicting a long-run count or share, not a single outcome?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I predicting a long-run count or share, not a single outcome?

Worked Examples

Example 1

easy
A basketball player makes free throws with probability 0.75. In 200 free throws, how many do we expect her to make?

Answer

Expected makes =200×0.75=150= 200 \times 0.75 = 150 free throws.

First step

1
Expected count formula: E=n×PE = n \times P

Full solution

  1. 2
    Substitute: E=200×0.75=150E = 200 \times 0.75 = 150
  2. 3
    Interpretation: on average, she will make 150 of 200 free throws
  3. 4
    Note: this is the long-run average — any single game of 200 shots might yield slightly more or fewer
Expected count =n×P= n \times P gives the average number of successes in nn trials with probability PP. This is the long-run mean of repeated experiments, not a guaranteed exact count for any single trial.

Example 2

medium
A game has three outcomes: win \$10 (prob 0.2), break even \$0 (prob 0.5), lose \$5 (prob 0.3). Calculate the expected value and interpret what it means for 1000 games.

Example 3

medium
A factory ships parts with defect rate 0.020.02. In a shipment of 50005000 parts, how many defective parts are expected?

Example 4

medium
Why doesn't the law of large numbers guarantee exactly 5050 heads in 100100 flips of a fair coin?

Example 5

hard
A game has expected value $0.50 per play. After 100 plays, you've won $10 (versus expected $50\$50). Is the game broken?

Example 6

hard
A fair coin is tossed 1010 times. Expected number of heads is 55. Is P(exactly 5 heads)=1P(\text{exactly 5 heads}) = 1?

Example 7

medium
Why are casinos profitable in the long run even if individual players can win big?

Example 8

hard
A school predicts 5%5\% of students will miss the bus. The school buses 400400 students. Expected riders and expected missers?

Example 9

challenge
You may buy a $5 ticket that pays $100 with probability 0.040.04. Should you buy?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A school expects 15% of students to be absent on any given day. If there are 300 students, how many absences are expected?

Example 2

hard
An insurance company charges \$200/year for a policy. It pays \$10,000 if the insured event occurs (probability 0.01) and \$0 otherwise. Calculate the insurance company's expected profit per policy.

Example 3

easy
P(heads)=0.5P(\text{heads})=0.5. In 200 flips, how many heads do you expect?

Example 4

easy
A die has P(6)=1/6P(6)=1/6. In 60 rolls, how many 6s are expected?

Example 5

easy
Probability is interpreted as the long-run relative frequency over many trials. True or false?

Example 6

easy
If P(rain)=0.2P(\text{rain})=0.2 each day, how many rainy days are expected in 30 days?

Example 7

easy
Does P=0.5P=0.5 guarantee exactly 50 heads in 100 flips?

Example 8

easy
A game pays $2 with probability 1/21/2 and $0 otherwise. What is the expected payout per play?

Example 9

easy
If P(defect)=0.01P(\text{defect})=0.01, how many defects are expected in 500 items?

Example 10

easy
After 5 straight losses in an independent game with P(win)=0.3P(\text{win})=0.3, what is the probability of winning the next round?

Example 11

medium
A die game pays the number rolled in dollars (1–6, fair). What is the expected payout per roll?

Example 12

medium
A lottery ticket costs $2 and wins $1000 with probability 1/10001/1000. What is the expected net gain per ticket?

Example 13

medium
In 100 flips of a fair coin you got 57 heads. Does this contradict P(heads)=0.5P(\text{heads})=0.5?

Example 14

medium
A spinner pays $10 on green (p=0.2p=0.2) and costs $3 otherwise. Expected value per spin?

Example 15

medium
Over 1000 trials of an event with P=0.1P=0.1, the expected count is 100. If you observe 87, what does the law of large numbers say about more trials?

Example 16

medium
A carnival game: roll two dice; win $5 if the sum is 7, else lose $1. P(sum=7)=6/36=1/6P(\text{sum}=7)=6/36=1/6. Expected value?

Example 17

medium
A factory expects 5 defects per 500 items (p=0.01p=0.01). One batch of 500 has 12 defects. Is this within normal expectation or a possible signal?

Example 18

medium
A weighted coin has P(H)=0.7P(\text{H})=0.7. In 50 flips, how many heads are expected, and how many tails?

Example 19

medium
A game gives +$3+\$3 with probability 0.40.4 and −$2-\$2 with probability 0.60.6. Over 100 plays, what total net result is expected?

Example 20

challenge
A bet wins $a with probability 0.250.25 and loses $2 with probability 0.750.75. For what value of aa is the game fair (expected value 0)?

Example 21

challenge
You draw one card from a standard 52-card deck; you win \$13 for an ace and \$0 otherwise, but pay \$1 to play. What is the expected net value?

Example 22

challenge
An insurer charges $200/year. A claim of $10000 occurs with probability 0.0150.015. What is the insurer's expected profit per policy?

Example 23

easy
A coin lands heads with probability 0.50.5. In 400400 flips, how many heads do you expect?

Example 24

easy
A die shows 33 with probability 1/61/6. In 300300 rolls, how many 33s are expected?

Example 25

medium
A game pays $3 with probability 0.40.4 and $0 otherwise. What is the expected payout per play?

Example 26

medium
A raffle ticket costs $2, has a 1/10001/1000 chance of winning $1000. Find the expected net gain per ticket.

Example 27

medium
A class of 2525 students has, on average, 4%4\% absentee rate. Expected absences per day?

Example 28

easy
A bag has 20%20\% red marbles. In 5050 draws with replacement, expected number of red marbles?

Example 29

medium
P(seven on two dice)=1/6P(\text{seven on two dice}) = 1/6. In 6060 rolls of two dice, expected number of 77s?

Example 30

hard
A roulette wheel has 3838 slots (1818 red, 1818 black, 22 green). A $1 bet on red pays $1 if red, −-$1 otherwise. Expected gain per spin?

Example 31

medium
An archer hits the bullseye 30%30\% of attempts. Expected bullseyes in 5050 shots?

Example 32

medium
80%80\% of customers buy item A. Expected buyers among 250250 customers?

Example 33

medium
A box has 100100 tickets: 55 marked $50, 2020 marked $5, rest $0. Expected value per draw?

Example 34

easy
A spinner shows red 40%40\% of the time. Expected red landings in 2020 spins?

Example 35

hard
An insurance policy pays out $50{,}000 with probability 0.0020.002, otherwise nothing. The premium is $120. Expected profit per policy?

Example 36

medium
A test has 44 multiple-choice questions with 44 options each. Random guessing — expected correct answers?

Example 37

medium
A coupon gives $5 off with probability 0.10.1. Expected savings per use?

Example 38

hard
A game pays $10 with probability 0.10.1, $1 with probability 0.50.5, −$5-\$5 otherwise. Expected value?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

probability