Probability paradoxes are fascinating concepts in mathematics that often defy human intuition. While the rules of probability are clear and logical, some situations yield results that seem counterintuitive. This article explores several famous probability paradoxes and explains why the outcomes are what they are.
The Monty Hall Problem is one of the most famous probability paradoxes. The problem is based on a quiz show, where the contestant must choose one of three doors. Behind one door is a car (the grand prize), while the other two doors hide goats. After the contestant chooses a door, the host opens one of the other two doors, revealing a goat. The contestant is then given the option to stick with their original choice or switch to the remaining door. The answer is to switch, as the chances of winning increase from 1/3 to 2/3. Our initial intuition is often wrong because we assume that after one door is opened, the chances become 50-50, which is not the case (Selvin, 1975; Granberg & Brown, 1995).
The Birthday Paradox demonstrates how quickly the probability of two people sharing a birthday in a small group increases. Mathematically, with just 23 people in a group, the probability that two people share the same birthday reaches 50%. This result seems strange because we often assume that a much larger group is required. However, the calculation shows that the number of pairs compared grows exponentially as the group size increases (Bolker, 2006).
Simpson’s Paradox occurs when trends that appear in several groups of data reverse or disappear when the data is aggregated. For example, a university with two programs may show higher acceptance rates for men in one program and higher rates for women in the other. However, when the data for both programs are combined, the overall acceptance rate for men is higher, even though this trend did not appear in the individual programs. This phenomenon happens because the distribution of applicants in each program affects the aggregate result (Pearl, 2009).
Gambler’s Fallacy is the belief that past random results influence the likelihood of future outcomes. An example is the belief that if a coin is flipped five times and always lands on heads, the next flip is more likely to land on tails. However, each coin flip is an independent event, so the probability remains 50% for each side (Tversky & Kahneman, 1971).
Probability paradoxes teach us to think critically and understand that intuition is often unreliable in probability situations. By studying these paradoxes, we can improve our logical thinking and make better decisions in situations involving uncertainty. Have you ever faced a situation where your intuition turned out to be wrong?
Keywords: Paradox, Mathematics, Probability
References:
- Selvin, S. (1975). A Problem in Probability (Letter to the Editor). The American Statistician, 29(1), 67.
- Granberg, D., & Brown, T. A. (1995). The Monty Hall Dilemma. Personality and Social Psychology Bulletin, 21(7), 711–723.
- Bolker, E. (2006). The Birthday Problem. The Mathematical Intelligencer, 28(4), 31–35.
- Pearl, J. (2009). Causality: Models, Reasoning, and Inference. Cambridge University Press.
- Tversky, A., & Kahneman, D. (1971). Belief in the Law of Small Numbers. Psychological Bulletin, 76(2), 105–110.
Author: Meilinda Roestiyana Dewy