A bet pays $500 with 10% probability and loses $40 with 90% probability. Expected value = $500(0.1) - $40(0.9) = +$14. You take this bet independently 100 times at small stakes. Which statement best describes the correct expected-value reasoning?
AThis is a bad bet because it loses 90% of the time — the high loss rate makes it inadvisable regardless of the payoff
BThis is a good bet in expectation; taken repeatedly at manageable stakes, the rare wins should more than offset the frequent small losses over time
CExpected value reasoning does not apply here because the outcomes are not equally likely
DThis is only a good bet on the first trial; after the first loss, you should stop because the 90% loss rate has 'used up' future bad luck
The expected value is +$14 per trial — a good bet despite losing on 90 of 100 attempts. Over 100 independent trials, you should expect roughly +$1,400 in total profit while losing ~90 individual bets. The key insight is that expected value is a long-run average, not a per-trial guarantee. Judging the bet by its loss probability alone (option A) misses the magnitude of the wins entirely. The 'gambler's fallacy' in option D is also wrong: each trial is independent. The practical condition for EV reasoning is that stakes are small enough that a long losing streak doesn't wipe out your ability to keep playing.
Question 2 Multiple Choice
A student argues: 'Since this investment has positive expected value, I should take it regardless of how large the potential loss is relative to my total savings.' What is the most important limitation this reasoning ignores?
APositive expected value calculations are only valid when the probabilities are known with certainty
BWhen stakes are large relative to your total resources, variance matters — a loss that eliminates your ability to make future decisions deserves more weight than raw expected value captures
CExpected value reasoning requires risk-neutrality, and since everyone is risk-averse, the calculation is always misleading
DThe student's reasoning is correct; rational agents should always maximize expected monetary value regardless of stake size
Expected value reasoning works best when you can make many decisions at similar stakes, allowing the law of large numbers to operate and variance to wash out. When a single bet could wipe out your financial foundation — eliminating your ability to make future positive-EV bets — the variance of outcomes is decision-relevant even if the EV is positive. This is the intuition behind the Kelly criterion and expected utility theory: the marginal utility of resources diminishes as wealth decreases, so losing everything is worse than the raw dollar amount suggests. Rational decision-making must account for this, especially at high stakes.
Question 3 True / False
A bet with positive expected value is expected to produce a positive outcome on any individual trial.
TTrue
FFalse
Answer: False
Expected value is a probability-weighted long-run average, not a per-trial guarantee. A coin flip paying +$3 on heads and -$1 on tails has an expected value of +$1, but on any single flip you either gain $3 or lose $1 — there is no 'average' outcome. The positive EV means that across many such flips, your average outcome per trial converges toward +$1 by the law of large numbers. Confusing expected value with guaranteed outcome is one of the most consequential errors in probabilistic reasoning — it leads people to either over-trust individual positive-EV bets or abandon correct strategies after a few unlucky trials.
Question 4 True / False
A bet that loses most of the time can still be the correct bet to take if the payoff when it wins is large enough to produce positive expected value.
TTrue
FFalse
Answer: True
This is the core of expected value reasoning. Venture capital investments, lottery-style payoffs, and many strategic opportunities involve bets that lose frequently but carry large enough upside to produce positive expected value. Evaluating bets purely by win probability (without weighting by magnitude of each outcome) systematically undervalues rare high-payoff opportunities and leads to excessive risk aversion. The correct calculation weights both the probability AND the magnitude of every possible outcome. A 5% chance of $1,000 has higher expected value than a 60% chance of $50, even though the first bet loses 95% of the time.
Question 5 Short Answer
Why does expected value reasoning say that systematically taking positive-EV bets produces better outcomes over time, even though any individual bet may lose?
Think about your answer, then reveal below.
Model answer: Expected value is a probability-weighted average across all possible outcomes. For any single trial, randomness determines the result, and even the best bet can lose. But the law of large numbers states that as the number of independent trials increases, the observed average outcome converges toward the theoretical expected value. If you consistently take positive-EV bets, your cumulative result trends upward over many decisions. Avoiding positive-EV bets out of loss aversion means leaving that expected value on the table every time. The key practical condition is that individual stakes remain small enough relative to total resources that you survive inevitable losing streaks long enough for the averages to work in your favor.
This is why the practical context of EV reasoning matters. A single investor who makes one high-variance bet and goes broke cannot benefit from the positive expected value — they exit the game before the law of large numbers operates. An investor who makes diversified positive-EV bets across many opportunities captures most of the theoretical return. EV reasoning is most powerful as a portfolio strategy rather than a one-shot framework, which is why the topic notes that variance matters when stakes are large relative to resources.