Is God a Mathematician_ - Mario Livio [66]
The essence of probability theory can be gleaned from the following simple facts. No one can predict with certainty which face a fair coin tossed into the air will show once it lands. Even if the coin has just come up heads ten times in a row, this does not improve our ability to predict with certainty the next toss by one iota. Yet we can predict with certainty that if you toss that coin ten million times, very close to half the tosses will show heads and very close to half will show tails. In fact, at the end of the nineteenth century, the statistician Karl Pearson had the patience to toss a coin 24,000 times. He obtained heads in 12,012 of the tosses. This is, in some sense, what probability theory is really all about. Probability theory provides us with accurate information about the collection of the results of a large number of experiments; it can never predict the result of any specific experiment. If an experiment can produce n possible outcomes, each one having the same chance of occurring, then the probability for each outcome is 1/n. If you roll a fair die, the probability of obtaining the number 4 is 1/6, because the die has six faces, and each face is an equally likely outcome. Suppose you rolled the die seven times in a row and each time you got a 4, what would be the probability of getting a 4 in the next throw? Probability theory gives a crystal-clear answer: The probability would still be 1/6—the die has no memory and any notions of a “hot hand” or of the next roll making up for the previous imbalance are only myths. What is true is that if you were to roll the die a million times, the results will average out and 4 would appear very close to one-sixth of the time.
Let’s examine a slightly more complex situation. Suppose you simultaneously toss three coins. What is the probability of getting two tails and one head? We can find the answer simply by listing all the possible outcomes. If we denote heads by “H” and tails by “T,” then there are eight possible outcomes: TTT, TTH, THT, THH, HTT, HTH, HHT, HHH. Of these, you can check that three are favorable to the event “two tails and one head.” Therefore, the probability for this event is 3/8. Or more generally, if out of n outcomes of equal chances, m are favorable to the event you are interested in, then the probability for that event to happen is m/n. Note that this means that the probability always takes a value between zero and one. If the event you are interested in is in fact impossible, then m = 0 (no outcome is favorable) and the probability would be zero. If, on the other hand, the event is absolutely certain, that means that all n events are favorable (m = n) and the probability is then simply n/n = 1. The results of the three coin tosses demonstrate yet another important result of probability theory—if you have several events that are entirely independent of each other, then the probability of all of them happening is the product of the individual probabilities. For instance, the probability of obtaining three heads is 1/8, which is the product of the three probabilities of obtaining heads in each of the three coins: 1/2 × 1/2 × 1/2 = 1/8.
OK, you may think, but other than in casino games and other gambling activities, what additional uses can we make of these very basic probability concepts? Believe it or not, these seemingly insignificant probability laws are at the heart of the modern study of genetics—the science of the inheritance of biological characteristics.
The person who brought probability into genetics was a Moravian priest. Gregor Mendel (1822–84) was born in a village near the border between Moravia and Silesia (today Hyncice in the Czech Republic). After entering the Augustinian Abbey of St. Thomas in Brno, he studied