The Misconception of Likelihood in Probabilistic Events

Expectations and probability are closely linked to our understanding of events. When an event has a 50% chance of happening, does its likelihood increase over time? The answer depends on how we interpret and apply probability. This article aims to clarify misconceptions around the likelihood of events with a 50% probability and explore why the individual probability of each event remains constant.

Understanding Probability and Expectation

If an event A has a 50% chance of occurring, this means that based on the available information, it is just as likely to happen as not. As new information becomes available, this probability may change. For example, if you can see a coin on a table, and you determine that it is not possible for it to be heads, then the probability of event A occurring changes.

Examples of Probability and Likelihood

Coin Tosses

The likelihood of a coin landing heads is 50%. If you toss a coin twice, there are four possible outcomes: HH, HT, TH, and TT. Since the only outcome that does not have at least one heads is TT, the probability of getting at least one heads in two attempts is not 75%. Instead, it is 75%, because three out of the four outcomes include at least one heads. However, each individual throw still has a 50% chance of being heads, independent of previous outcomes.

Multiple Coin Tosses

If you toss a coin ten times, the likelihood of at least one heads does increase somewhat, since there are 1,024 possible sequences, and only one of them (TTTTTTTTTT) has no heads. Despite this, each individual toss maintains a 50% chance of being heads.

Radioactive Decay

Similar principles apply to radioactive decay. If an element has a half-life of one year, half of the atoms in a sample will decay in that year. The probability of any specific atom decaying in the following year remains 50%, regardless of whether it has already decayed.

Survival Probability

In another scenario, a newborn baby girl in a developed country has a 50% chance of living to her 88th birthday. This is a statistical probability based on large samples. However, if a person is already 88, their chances of surviving another year are much higher than 50%. This is a testament to the fact that the probability of what happens next is not a function of what has happened so far.

Mathematical vs. Experimental Probability

It is important to distinguish between mathematical and experimental probabilities. Mathematical probability is based on theoretical models, while experimental probability is based on observable data.

In the above examples, even if the experimental outcomes (like a candidate never getting elected) differ from the mathematical expectations, there is always a bias or underlying reason. In the case of the candidate, the bias is the wishes of the electorate, which may influence the outcome.

Conclusion

The likelihood of events with a 50% probability does not increase over time. Each event has an independent 50% probability, irrespective of past outcomes. Understanding this distinction is crucial for making informed decisions and not falling into the trap of believing in a 'growing' likelihood to change the game.

By grasping these fundamental concepts, you can make more accurate predictions and avoid the pitfalls of probabilistic fallacies.