The gambler's fallacy

• A. The belief that past random events influence the probability of future independent events
• B. The belief that independent random events become more or less likely based on what has happened previously ✓
• C. The tendency to bet more money after a series of wins
• D. The assumption that casinos always win in the long run

Why: The gambler's fallacy is the mistaken belief that independent random events are influenced by previous outcomes, so that a streak of one result makes the opposite result 'due' to occur.

• A. Less than 50%, because tails is due
• B. Greater than 50%, because the streak is likely to continue
• C. Exactly 50%, because each flip is independent ✓
• D. About 3%, because six heads in a row is very unlikely

Why: Each coin flip is independent, so the probability of heads on any single flip remains 50% regardless of previous results. The fallacy lies in believing the past outcomes change this probability.

• A. Believing a roulette wheel that has landed on black 8 times is more likely to land on red next ✓
• B. Calculating the odds of rolling a six on a fair die
• C. Recognizing that a lottery ticket has a very small chance of winning
• D. Understanding that a card deck is reshuffled before each game

Why: Thinking red is 'due' after a run of blacks is the gambler's fallacy, because each spin of a fair roulette wheel is independent of previous spins.

• A. They misunderstand that random sequences should look balanced even in the short term ✓
• B. They have perfect knowledge of the probabilities involved
• C. They believe that dice and coins have memories
• D. They understand that the law of large numbers applies to single events

Why: People expect short random sequences to look representative of the overall probability, so a streak feels like it must be corrected soon. In reality, randomness does not self-correct over short runs.

• A. It guarantees that a streak will end after a fixed number of trials
• B. It states that over many trials, the observed proportion tends toward the expected probability, without requiring any single outcome to compensate for earlier ones ✓
• C. It proves that past outcomes determine future results in games of chance
• D. It shows that short-term streaks are impossible in random processes

Why: The law of large numbers says the average result converges to the expected value over many trials, but it does not force individual outcomes to balance out. The fallacy confuses these two ideas.

• A. Less than 50%, because tails is due
• B. 50%, because each flip is independent ✓
• C. Greater than 50%, because the streak is likely to continue
• D. Cannot be determined without more data

Why: Coin flips are independent events. The coin has no memory, so the probability of heads on any single flip remains 1/2 regardless of previous outcomes. Believing tails is 'due' is the gambler's fallacy.

• A. Red is more likely because black has been overdue to lose
• B. Black is more likely because the wheel is 'hot'
• C. Red and black are equally likely, since each spin is independent ✓
• D. Red is guaranteed to win soon because of the law of averages

Why: Each spin of a fair roulette wheel is independent of previous spins. The chance of red (on a European wheel, 18 of 37 pockets) is the same on every spin. The 'law of averages' does not force short-term outcomes to balance out.

• A. She is correct, because over time the coin must balance out on each individual flip
• B. She confuses the law of large numbers, which concerns long-run proportions, with a rule that short sequences must self-correct ✓
• C. She should expect tails again because coins are biased toward streaks
• D. She miscalculated the probability of nine tails in a row

Why: The law of large numbers says the proportion of heads will approach 1/2 over many flips. It does not mean the coin remembers past results and compensates on the next flip. The short-term streak does not change the odds of the next flip.

• A. Higher than usual, because she is due to make one
• B. Lower than usual, because she is in a slump that is likely to continue
• C. The same as her usual success rate, because past misses don't change the chance of the next shot ✓
• D. Zero, because a streak of misses means she has lost her ability

Why: If shots are independent and skill is constant, each attempt has the same success probability as always. A run of misses is not evidence the next one will fail or succeed, though real changes in skill (like injury) would be a separate matter.

• A. Believing that after several heads in a row, tails is more likely on the next flip of a fair coin ✓
• B. Calculating that the chance of rolling two sixes with two dice is 1 in 36
• C. Recognizing that a fair die has a 1/6 chance of landing on any face
• D. Understanding that the probability of drawing an ace from a fresh deck is 4/52

Why: The gambler's fallacy is the mistaken belief that a random process is 'due' to reverse after a run of the same outcome. The other options correctly apply probability to independent events and do not involve the faulty expectation that past results influence future ones.

• A. Less than 50%, because tails is 'due'
• B. 50%, because each flip is independent ✓
• C. Greater than 50%, because heads is on a streak
• D. 0%, because the sequence must balance out immediately

Why: Each flip of a fair coin is independent. Past outcomes do not change the probability of future outcomes, so the chance of heads remains 50%. Believing tails is 'due' is the gambler's fallacy.

• A. Short-term results will quickly correct themselves to match the expected probability
• B. As the number of trials increases, the observed proportion tends to approach the expected probability ✓
• C. A run of bad luck makes good luck more likely in the next trial
• D. Random events become predictable once enough data is collected about them

Why: The Law of Large Numbers states that the average of results from many trials converges toward the expected value. It says nothing about short-term results correcting themselves, which is a common misreading.

• A. Confusing independent events with dependent events ✓
• B. Applying the Law of Large Numbers to a single short sequence
• C. Assuming the wheel is rigged
• D. Ignoring the house edge entirely

Why: Each spin of a fair roulette wheel is independent of previous spins. The gambler treats past black results as affecting the chance of red, which is the gambler's fallacy, a mistake of treating independent events as dependent.

• A. 'The odds are even this time.'
• B. 'The law of averages says things must even out soon.' ✓
• C. 'Each trial is a fresh start.'
• D. 'The sample size is too small to draw conclusions.'

Why: 'The law of averages says things must even out soon' reflects the misconception that short-term deviations are corrected by later outcomes. The correct idea is that imbalances are diluted over many trials, not repaid by future ones.

• A. It is higher than 50%, since a girl is now overdue
• B. It is lower than 50%, since the family's pattern is unusual
• C. It is about 50%, assuming births are independent and the chance of a girl is roughly even ✓
• D. It is near 0%, since the sex ratio must rebalance within the family

Why: Assuming independent births with roughly equal odds for each sex, each birth has about a 50% chance of being a girl regardless of previous births. Past births do not create a debt that the next birth must repay.

• A. The hot hand fallacy
• B. The gambler's fallacy ✓
• C. The sunk cost fallacy
• D. The availability heuristic

Why: The gambler's fallacy is the mistaken belief that past independent outcomes change the probability of future ones. Each roulette spin is independent, so black still has the same chance (about 47.4% on a American wheel with 0 and 00) regardless of the previous streak.

• A. Greater than 50%, because tails is overdue
• B. Less than 50%, because the streak is unlikely to continue
• C. Exactly 50%, because each flip is independent ✓
• D. Exactly 1/64, because that is the probability of seven heads in a row

Why: Each flip of a fair coin is independent, so the probability of tails on any single flip remains 50%. The earlier heads do not make tails more likely. The 1/64 figure describes the chance of the whole sequence of seven heads, not the next flip.

• A. Past losses create a debt that must be repaid by future wins
• B. Coin tosses are not truly random, so streaks should be expected
• C. Each toss is independent, so the previous losses do not change the odds of winning the next toss ✓
• D. Five losses is too small a sample to say anything about luck

Why: The gambler's fallacy treats independent events as if they were linked. Because each coin toss is independent, the five previous losses have no effect on the outcome of the next toss, which remains 50/50.

• A. Stock prices always follow trends, so the decline should continue
• B. Short-term price moves are often close to random, so a losing streak does not guarantee an upcoming rise ✓
• C. Investors should never buy stocks that have declined
• D. The investor has ignored the company's dividend history

Why: While markets can have patterns and reasons for movement, treating a streak of declines as a guarantee of a coming rebound is the gambler's fallacy. Daily price changes are largely unpredictable from past streaks alone, so the expectation of a rebound is not justified by the streak itself.

• A. The fan is right, because batters have natural cycles of success
• B. The fan is right, because the pitcher will tire after several strikeouts
• C. The fan is wrong, because each at-bat is roughly independent and the player's chance of a hit does not increase after failures ✓
• D. The fan is wrong, because strikeouts are caused only by bad luck

Why: Treating a run of failures as making success more likely is the gambler's fallacy. Although real factors like fatigue or form can matter, the mere fact of prior strikeouts does not raise the probability of a hit on the next at-bat.

• A. Less than 50%, because tails is now due
• B. Exactly 50%, because each flip is independent ✓
• C. Greater than 50%, because heads is on a streak
• D. Close to 0%, because the pattern must break soon

Why: Each coin flip is independent, so past outcomes do not change the probability of the next flip. The chance of heads remains 50%. Believing tails is 'due' is the gambler's fallacy.

• A. A sound application of the law of large numbers
• B. An example of the gambler's fallacy ✓
• C. A correct use of Bayesian updating
• D. A reasonable strategy given the wheel may be biased

Why: The gambler assumes that a run of black makes red more likely on the next spin. Each spin is independent, and the wheel has no memory, so this reasoning is the gambler's fallacy.

• A. People expect short sequences of random outcomes to look like the overall long-run distribution ✓
• B. People know that random events are always perfectly balanced
• C. People overestimate how much their own skill affects chance outcomes
• D. People can accurately calculate probabilities for independent events

Why: The representativeness heuristic leads people to expect that small samples will closely resemble the overall probability pattern, such as heads and tails alternating evenly. This makes a streak seem like it must be corrected soon.

• A. The coin must eventually compensate for earlier results with a matching run of the opposite outcome
• B. The proportion converges because the average evens out over many trials, not because individual flips are influenced by past flips ✓
• C. Each flip becomes less random as the sequence grows longer
• D. The law of averages forces each individual outcome to move toward 50%

Why: The law of large numbers describes the overall proportion across many trials, which gradually approaches the expected value. It does not mean that past results cause future results to correct. Early streaks are diluted by the growing number of trials, not reversed.

• A. The fan is right, since streaks of misses are always followed by makes
• B. The fan is right only if the player is tired
• C. The fan is committing the gambler's fallacy, unless the misses reflect a real change in skill or conditions ✓
• D. The fan is right, because a player's skill is always reset after a miss

Why: If the misses are independent random events from a stable skill level, the next shot's chance is unchanged, so 'due' has no basis. However, if the misses reveal a real change such as an injury or fatigue, updating the estimate is reasonable. The fallacy lies in assuming a correction is owed by chance alone.

• A. Less than 50%, because tails is now 'due'
• B. Exactly 50%, because each flip is independent ✓
• C. More than 50%, because the coin is on a streak
• D. Close to 0%, because the streak must break

Why: Each coin flip is independent of previous flips. A fair coin has no memory, so the probability of heads on the next flip remains 50%. Believing tails is 'due' is the gambler's fallacy.

• A. Confusing independent events with dependent events ✓
• B. Ignoring the house edge on the wheel
• C. Assuming the wheel is biased toward black
• D. Misunderstanding the law of large numbers

Why: Each spin of a fair roulette wheel is independent. Past outcomes do not change future probabilities, so treating red as 'overdue' wrongly treats independent events as if they were dependent.

• A. Short streaks will be balanced out by the next few outcomes
• B. Over many trials, the proportion of outcomes approaches the expected probability, but this does not require corrective streaks ✓
• C. Every streak of losses guarantees a future streak of wins
• D. Large samples always show perfectly even results

Why: The law of large numbers says the observed proportion converges toward the true probability over many trials. It does not mean that a deviation is corrected by a compensating run in the short term; the imbalance is simply diluted as the sample grows.

• A. He is more likely to make the next shot because his shooting percentage is 75% overall
• B. He is due to make the next shot because his misses have to even out ✓
• C. His coach should review his shooting form
• D. The sample of 5 misses is too small to judge his skill

Why: Saying he is 'due' to make a shot because the misses must even out assumes his past misses influence future results. Independent attempts do not work that way, so this is the gambler's fallacy in action.

• A. Less than 50%, since a girl is now due
• B. About 50% ✓
• C. More than 50%, since the family may be prone to girls
• D. Close to 100%

Why: Assuming independent births with equal odds, each birth has a 50% chance of being a girl regardless of the previous five. Believing a girl is 'due' is the gambler's fallacy.

• A. Less than 50%, because tails is now due
• B. 50%, because each flip is independent of previous flips ✓
• C. Greater than 50%, because heads is on a streak
• D. Close to 0%, because the streak must end soon

Why: Each coin flip is independent. The coin has no memory, so the probability of heads remains 50% regardless of past results. Believing tails is 'due' is the gambler's fallacy.

• A. Believing that past random events change the probability of future independent random events ✓
• B. Believing that large sample sizes produce more reliable results
• C. Believing that skilled players can consistently beat games of pure chance
• D. Believing that rare events are impossible

Why: The gambler's fallacy is the mistaken belief that past outcomes of independent random events affect future outcomes, such as thinking a roulette ball is more likely to land on black after several reds.

• A. Your friend is right, because outcomes balance out quickly
• B. Your friend is right, because the wheel must be biased toward black
• C. Your friend is wrong, because each spin is independent and black still has the same chance as before ✓
• D. Your friend is right, but only if the wheel is old

Why: Each spin of a fair wheel is independent. Black's probability on any single spin remains the same regardless of previous results. Outcomes do not balance out on a short-term basis; they only average out over a very large number of trials.

• A. Because they understand probability theory very well
• B. Because they expect short sequences of random outcomes to look 'representative' and balanced ✓
• C. Because casinos deliberately rig outcomes against streaks
• D. Because they assume the law of large numbers works in the short term

Why: People expect random sequences to look balanced even in short runs. When a sequence looks too one-sided, they feel the other outcome must come next to restore balance, even though randomness doesn't work that way. Option D is a related misconception, but the core cognitive driver is the expectation of representativeness.

• A. A girl is more likely because the family is statistically due for one
• B. A girl is less likely because boys run in families
• C. The chance of a girl is still about 50%, since previous births do not affect the next birth ✓
• D. The chance of a girl is about 80% because of the four-boy streak

Why: Assuming independent births with roughly equal odds, each birth has about a 50% chance of being a girl regardless of earlier children. Past births do not create a debt that the next birth must repay.