👀 Probabilistic paradoxes

Gambler's Fallacy

The gambler’s fallacy is a cognitive bias in which a person believes that past random events influence the probability of future ones, provided the events appear "connected" within a sequence. It is one of the most common pitfalls, both in gambling and in everyday decision-making.

Example of a coin:

Suppose a coin lands on heads 5 times in a row. Many people might think, "Now, tails is more likely to come up, since it hasn't appeared in a while." That is not the case. The probability of getting tails on each new flip remains strictly 50%, regardless of what happened before. A coin has no memory.

Mathematically speaking, for a fair coin, \(P(\text{heads}) = 0.5\) at all times, and the events are independent. Even after 100 consecutive heads, the probability of getting tails on the 101st flip remains 50%.

The law of large numbers

The law of large numbers states that as the number of coin flips increases, the proportion of heads tends toward 50%. However, this does not mean that past deviations will be canceled out. Imagine that after 5 consecutive heads, you perform another 1,000 flips. The final proportion of heads will be approximately 50% (for instance, 502 heads and 498 tails (50.2%)). The absolute "excess" of 5 heads has not disappeared, but against the backdrop of 1,005 total flips, it becomes statistically insignificant. The law operates by "drowning out" past deviations with new data, rather than by compensating for them.

Monty Hall paradox

One of the most famous paradoxes illustrating how intuition can lead us astray when estimating probabilities. Imagine a game show: there are three doors in front of you. Behind one is a car, and behind the other two are goats. You choose a door (for example, door No.1). The host, who knows where the car is, opens another door (let’s say, door No.3) to reveal a goat. He then offers you the chance to switch your choice to the remaining door, No.2. Should you change your decision?

Intuition suggests that since two doors remain, the odds are 50/50, meaning it makes no difference. However, the correct answer is that switching doors is advantageous — the probability of winning by switching is \( \frac{2}{3} \), whereas sticking with your initial choice yields a probability of only \( \frac{1}{3} \).

Why? Initially, the probability of choosing the car is \( \frac{1}{3} \), while the probability of choosing a goat is \( \frac{2}{3} \). If you initially pick a goat (a \( \frac{2}{3} \) probability), then once the host reveals another goat, the remaining door is guaranteed to lead to the car. By switching your choice, you win in those \( \frac{2}{3} \) of cases. If, however, you initially chose the car (a \( \frac{1}{3} \) probability), then switching will result in a loss. Thus, switching doors gives you twice the chance of winning.

This paradox is directly applicable to games involving choice and hidden information (for example, card games or the strategies used in the show "Who Wants to Be a Millionaire?").

Birthday paradox

How many people need to be in a room for the probability of a shared birthday (the same day and month, ignoring the year) to be greater than 50%? Intuitively, it might seem like a large number, around 183 (half of 365). In reality, just 23 people are enough. The probability that at least two out of 23 people share a birthday exceeds 50%.

Formula: \( P(\text{match}) = 1 - \frac{365}{365} \cdot \frac{364}{365} \cdots \frac{365-n+1}{365} \). For \(n=23\), this is approximately 0.507, while for \(n=50\), it reaches 0.97. This is significant in the context of gaming: for instance, in multiplayer online games, when rare items are distributed randomly, the occurrence of "lucky" coincidences happens far more often than one might expect. It also explains why, within groups of players, one frequently encounters individuals who happen to have identical "randomly" chosen nicknames or dates.

Two envelopes problem

A person is given two envelopes, one of which contains twice as much money as the other. They open one envelope and find \(X\) dollars inside. The host offers to let them swap their envelope for the other one. Should they do it? It might seem that the second envelope is equally likely to contain \(X/2\) or \(2X\), making the expected value of swapping \( \frac{1}{2} \cdot \frac{X}{2} + \frac{1}{2} \cdot 2X = 1.25X \), which is greater than \(X\). Does this mean that swapping is always advantageous? But if that were the case, after swapping, they would want to swap back, leading to a contradiction.

The paradox is resolved by the fact that the prior distribution of the amounts in the envelopes is not defined. If you do not know how the amounts were chosen, then the conditional probability \(P(\text{second envelope has more})\) is not necessarily equal to 1/2. In real-world scenarios (such as game shows involving envelopes), the organizers may specify a distribution, in which case the expected gain from switching is not always positive. This paradox serves as a lesson to be cautious when using expected value in situations where complete information about the distribution is lacking.

Law of streaks and Hot hand

Basketball players believe in the "hot hand" — the idea that after a few successful shots, the probability of the next one going in is higher. Statistics, however, often show that sequences of successes and failures follow the laws of chance. Long streaks (for example, 10 reds in a row at roulette) are inevitable. With a probability of \((18/37)^{10} \approx 0.07\%\), they are rare but real occurrences when a large number of trials take place. Players tend to perceive such streaks as a "pattern" and adjust their bets accordingly, a behavior that only serves to increase the casino's advantage.

Expected value and Subjective utility

The St. Petersburg paradox involves a game in which a coin is tossed repeatedly until the first "tails" appears. The payout is \(2^k\) dollars, where \(k\) is the number of the toss (i.e., 2 dollars if tails appears on the first toss, 4 dollars on the second, 8 on the third, and so on). The EV of the winnings is infinite: \(\sum_{k=1}^\infty \frac{1}{2^k} \cdot 2^k = \sum_{k=1}^\infty 1 = \infty\). Nevertheless, no one is willing to pay a large sum of money to participate in such a game, because the marginal utility of money diminishes and real-world individuals cannot risk an infinite amount. This demonstrates that when making decisions in games (especially those involving lotteries or high-risk strategies) one must consider not only the expected value but also one's own risk tolerance (or utility function).

Practical takeaway for players

Probabilistic paradoxes help us:

• Not rely on intuition when evaluating random events.
• Remember that independent events have no memory.
• When hidden information is present, recalculate probabilities by taking the host's actions into account.
• Interpret long streaks with caution — they are a natural feature of randomness.
• Use expected value as a tool but adjust it according to your own risk tolerance.

Understanding these paradoxes helps in making more rational decisions in board games, poker and video games involving elements of chance and even in real life.