Raising the bar

Chapter 4. Probability

Unit 4.5. Ways to find probabilities

Equally likely outcomes (theoretical approach).

(1) In case of a fair coin, the head and tail are equally likely, so $latex P(H) = P(T)$. Since these events are mutually exclusive and collectively exhaustive, by additivity and completeness axiom $latex P(H) + P(T) = 1$. It follows that $latex P(H) = P(T) = 1/2$. We have rigorously derived the coin part of Table 4.3 (see Probability).

(2) In case of a fair die the events $latex {1},...,{6}$ are equally likely, so they all have the same probability $latex p$. By additivity and completeness axiom $latex 6p = 1$, so $latex p = 1/6$. We have derived the die part of Table 4.3.

(3) More generally, suppose that there are $latex N$ basic outcomes and suppose the basic outcomes are equally likely. Then, as above, the probability of one basic outcome is $latex 1/N$. Now consider some condition that separates an event $latex A \subset S$ (in case of a die the condition could be that the outcome is even). Denote $latex {N_A}$ the number of basic outcomes satisfying the condition. Then by additivity

$latex P(A) = \underbrace {\frac{1}{N} + ... + \frac{1}{N}}_{{N_A}{\rm{\ times}}} = \frac{{{N_A}}}{N}.$

This formula is called classical probability in NCT. Many students remember the formula but don't think about the conditions under which it is true. Reread the above derivation and state the result in the form: if ... then ...

Frequentist approach. After observing relative frequencies in an experiment, one can take those frequencies as probabilities. The larger the sample size, the better will be the approximation to the truth. This is a general statistical fact. Statistics is mostly about large sample properties.

Expert opinions. When the theoretical and frequentist approaches are not applicable, one can use subjective probabilities assigned by experts. For example, performance of economic projects depends on overall macroeconomic conditions. In project evaluation the resulting statements sound like this: with probability 0.6 the economy will be on the rise, and then the profitability of the project will be this much and so on.

Exercise 4.3. We are going to simulate a large number of tossings of a fair coin. To work with numbers instead of heads and tails, we introduce a numerical variable

$latex C = 1\ \textrm{if head}, C=0\ \textrm{if tail}.$

In a large number of trials $latex n$ we calculate the proportion $latex {p_n}$ of heads. As $latex n$ increases, $latex {p_n}$ should approach $latex 1/2$.

Step 1. In cell A1 put the formula =IF(RAND()＞1/2,0,1). Copy it to cells A2-A100. These will be observations on the coin.

Step 2. In B1 type =SUM($A$1:A1)/ROW(A1) and copy this formula to B2-B100. The result in, say, B100 will be =SUM($A$1:A100)/ROW(A100). This is the sum of the contents of the cells A1 through A100 divided by the number of these cells. Thus, in cells B1-B100 we have averages of the first $latex n$ observations, $latex n = 1,...,100$.

Step 3. Select column B and see on the line plot the behavior of averages. They approach $latex 1/2$. To see the picture better, on the plot right-click the vertical axis and select "Format Axis". In Axis Options for "Minimum" check "Fixed" and enter 0 for the value, for "Maximum" check "Fixed" and enter 1 for the value, for "Major Unit" check "Fixed" and enter 0.1 for the value. Click OK. By pressing F9 you will change realizations of the means for sample sizes $latex n \le 100$. For low $latex n$, the mean can be far from $latex 1/2$ but it approaches $latex 1/2$ as $latex n$ increases. (In theory, this phenomenon is called the Law of Large Numbers. Gerolamo Cardano formulated it without a proof in the 16th century, Wikipedia: Law of large numbers.) Sometimes the speed of convergence of averages to $latex 1/2$ is not very good. This is a general probabilistic fact (see Unit 8.2 for the theory).

Exercise 4.4. Prove the addition rule $latex P(A \cup B) = P(A) + P(B) - P(A \cap B)$. Generalize it to the case of three events $latex P(A \cup B \cup C) = ...$ What is the difference between additivity and addition rule?

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