Practice Probability as Expectation in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Probability can be interpreted as the long-run relative frequency of an event over infinitely many identical trials of a random experiment.
means if you flip many times, about half will be heads.
Showing a random 20 of 50 problems.
Example 1
mediumA lottery ticket costs $2 and wins $1000 with probability . What is the expected net gain per ticket?
Example 2
mediumA factory expects 5 defects per 500 items (). One batch of 500 has 12 defects. Is this within normal expectation or a possible signal?
Example 3
easyA game pays $2 with probability and $0 otherwise. What is the expected payout per play?
Example 4
hardA game has expected value $0.50 per play. After 100 plays, you've won $10 (versus expected ). Is the game broken?
Example 5
mediumWhy doesn't the law of large numbers guarantee exactly heads in flips of a fair coin?
Example 6
mediumIn 100 flips of a fair coin you got 57 heads. Does this contradict ?
Example 7
easyIf , how many defects are expected in 500 items?
Example 8
medium of customers buy item A. Expected buyers among customers?
Example 9
mediumWhy are casinos profitable in the long run even if individual players can win big?
Example 10
easyFill in the blank: As trials increase, the relative frequency of an event tends toward its ___.
Example 11
mediumAn archer hits the bullseye of attempts. Expected bullseyes in shots?
Example 12
hardA roulette wheel has slots ( red, black, green). A $1 bet on red pays $1 if red, $1 otherwise. Expected gain per spin?
Example 13
easyA coin lands heads with probability . In flips, how many heads do you expect?
Example 14
mediumOver 1000 trials of an event with , the expected count is 100. If you observe 87, what does the law of large numbers say about more trials?
Example 15
hardA game pays $10 with probability , $1 with probability , otherwise. Expected value?
Example 16
mediumA carnival game: roll two dice; win $5 if the sum is 7, else lose $1. . Expected value?
Example 17
mediumA factory ships parts with defect rate . In a shipment of parts, how many defective parts are expected?
Example 18
easyA die has . In 60 rolls, how many 6s are expected?
Example 19
mediumA test has multiple-choice questions with options each. Random guessing — expected correct answers?
Example 20
mediumA 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.