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Mathematics for science and technology
Mathematics for science and technology

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3 Probability and common sense

The concept of probability is a purely theoretical one. Strictly speaking, no experiment measures a probability: all that you can measure is the fraction of times a particular outcome occurs in a finite number of attempts. In the infinitely long run this fraction is expected to approach the theoretical probability, but in practice you may never attain this limit. You could easily toss a fair coin four times and get four heads. You could even toss it 20 times and still get heads on every single toss, though that would be fairly unlikely. But the more tosses you made the more nearly the fraction

number of heads divided by total number of tosseswould approach its theoretical value of one divided by two.

A failure to appreciate the fact that the number of attempts needs to be extremely large before the probability of a particular outcome will reliably approach the theoretical value is at the root of many popular misconceptions about probabilities. One commonly held fallacy about coin tossing is that if the first ten tosses of a coin have produced several more heads than tails, then the eleventh toss is more likely than not to come up tails. This is not true. Although in the extremely long run the imbalance between heads and tails is expected to be negligible, on any one toss heads and tails are equally likely, irrespective of previous history. Coins have no memory!