**Properties of Outcomes—Law of Large Numbers**

Here are some formulas that describe properties of outcomes in various types of situations. Don’t let the symbology intimidate you.

**Law of Large Numbers**

Suppose you toss an “unweighted” die many times. You get numbers turning up, apparently at random, from the set {1, 2, 3, 4, 5, 6}. What will the average value be? For example, if you toss the die 100 times, total up the numbers on the faces, and then divide by 100, what will you get? Call this number *d* (for die). It is reasonable to suppose that *d* will be fairly close to the mean, *μ* :

*d* ≈ *μ* *d* ≈ (1 + 2 + 3 + 4 + 5 + 6)/6

= 21/6 = 3.5

It’s possible, in fact likely, that if you toss a die 100 times you’ll get a value of *d* that is slightly more or less than 3.5. This is to be expected because of “reality imperfection.” But now imagine tossing the die 1000 times, or 100,000 times, or even 100,000,000 times! The “reality imperfections” will be smoothed out by the fact that the number of tosses is so huge. The value of *d* will converge to 3.5. As the number of tosses increases without limit, the value of *d* will get closer and closer to 3.5, because the opportunity for repeated coincidences biasing the result will get smaller and smaller.

The foregoing scenario is an example of the *law of large numbers* . In a general, informal way, it can be stated like this: “As the number of events in an experiment increases, the average value of the outcome approaches the mean.” This is one of the most important laws in all of probability theory.

**Independent Outcomes**

Two outcomes *H* _{1} and *H* _{2} are *independent* if and only if the occurrence of one does not affect the probability that the other will occur. We write it this way:

*p* ( *H* _{1} ∩ *H* _{2} )= *p* ( *H* _{1} ) *p* ( *H* _{2} )

Figure 8-1 illustrates this situation in the form of a Venn diagram. The intersection is shown by the darkly shaded region.

**Fig. 8-1.** Venn diagram showing intersection.

A good example of independent outcomes is the tossing of a penny and a nickel. The face (“heads” or “tails”) that turns up on the penny has no effect on the face (“heads” or “tails”) that turns up on the nickel. It does not matter whether the two coins are tossed at the same time or at different times. They never interact with each other.

To illustrate how the above formula works in this situation, let *p* ( *P* ) represent the probability that the penny turns up “heads” when a penny and a nickel are both tossed once. Clearly, *p* ( *P* ) = 0.5 (1 in 2). Let *p* ( *N* ) represent the probability that the nickel turns up “heads” in the same scenario. It’s obvious that *p* ( *N* ) = 0.5 (also 1 in 2). The probability that both coins turn up “heads” is, as you should be able to guess, 1 in 4, or 0.25. The above formula states it this way, where the intersection symbol ∩ can be translated as “and”:

*p* ( *P* ∩ *N* ) = *p* ( *P* ) *p* ( *N* )

= 0.5 × 0.5 = 0.25

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