A probability expresses the observed frequency of the occurrence of a well-defined event for a large number of repetitions of the event, where each repetition is independent of the others (i.e., random). Thus the probability that a fair coin will come up heads in, say, 100 tosses is approximately 0.5; that is, it will come up heads approximately 50 percent of the time. (In the penultimate paragraph of this post, I explain why I emphasize approximately.)
If a coin is tossed 100 times, what is the probability that it will come up heads on the 101st toss? There is no probability for that event because it hasn’t occurred yet. The coin will come up heads or tails, and that’s all that can be said about it.
Scott Adams, writing about the probability of being killed by an immigrant, puts it this way:
The idea that we can predict the future based on the past is one of our most persistent illusions. It isn’t rational (for the vast majority of situations) and it doesn’t match our observations. But we think it does.
The big problem is that we have lots of history from which to cherry-pick our predictions about the future. The only reason history repeats is because there is so much of it. Everything that happens today is bound to remind us of something that happened before, simply because lots of stuff happened before, and our minds are drawn to analogies.
…If you can rigorously control the variables of your experiment, you can expect the same outcomes almost every time [emphasis added].
You can expect a given outcome (e.g., heads) to occur approximately 50 percent of the time if you toss a coin a lot of times. But you won’t know the actual frequency (probability) until you measure it; that is, after the fact.
Here’s why. The statement that heads has a probability of 50 percent is a mathematical approximation, given that there are only two possible outcomes of a coin toss: heads or tails. While writing this post I used the RANDBETWEEN function of Excel 2016 to simulate ten 100-toss games of heads or tails, with the following results (number of heads per game): 55, 49, 49, 43, 43, 54, 47, 47, 53, 52. Not a single game yielded exactly 50 heads, and heads came up 492 times (not 500) in 1,000 tosses.
What is the point of a probability statement? What is it good for? It lets you know what to expect over the long run, for a large number of repetitions of a strictly defined event. Change the definition of the event, even slightly, and you can “probably” kiss its probability goodbye.
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Fooled by Non-Randomness
Randomness Is Over-Rated
Beware the Rare Event
Some Thoughts about Probability
My War on the Misuse of Probability
Understanding Probability: Pascal’s Wager and Catastrophic Global Warming