obabilities p(heads), p(tails) as the number of trials increases. As expected, the counts for heads and tails are seemingly equal, at least when considering large numbers. However, the detail shows that time and again, the counts significantly depart from each other, meaning that there are more heads than tails or the reverse. But eventually these discrepancies seem to correct 1.1 Events, event space, and probabilities Heads Tails Number of trials Heads Tails Number of trials Figure 1.3 Experimental determination of the probability distribution of coin flipping, by means of 700 successive trials: (a) cumulative count of head and tails outcomes with inset showing detail for the first 100 trials, (b) corresponding probabilities. themselves as the game progresses, as if the coin would "know" how to come back to the 50:50 odds rule. Strange isn't it? The discrepancies between counts are reflected by the wide oscillations of the PDF (Fig. 1.3(b)). But as the experiment progresses, the oscillations are damped to the point where p(heads) 2 p(tails) 2 0.5, following an asymptotic behavior.2 Performing this experiment and obtaining such results is not straightforward. Different types of coins must be tested first, some being easier to flip than others, because of size or mass. Some coins seem not to lead Probability basics The above experiment illustrates the difference between event probabilities and their actual outcomes in the physical world. The nice thing about probability theory is that the PDF gives one a sense of the unknown when it comes to a relatively large number of outcomes, as if the unknown, or "chance," were domesticated by underlying mythical principles. On the other hand, a known probability gives no clue about a single event, just a sense of what it is most likely to be. A fair way to conceive of a 10% odds is that the corresponding event "should be observed" at least once after ten outcomes, and at least ten times after 100 outcomes, and very close to Q/10 times after Q outcomes, the closeness being in