Raising the bar

Chapter 4. Probability

Unit 4.7. All about joint events

The notions introduced here are interrelated but in NCT they are given in different places and without discussing the relationships between them. The discussion will be long, and if you are stuck with something, read everything from the start.

Example 4.5. In Kazakhstan the profits of wheat producers critically depend on weather conditions during summer (a lot of rain results in an abundant crop; there is always a lot of sun) and the weather during the harvesting time (just three rainy days in September are enough for the quality of wheat to plunge). Let us denote by $latex S$ the event that the weather is good in summer and by $latex H$ the event that it is good during the harvesting time. The farmers are happy when $latex P(S \cap H)$ is high.

Joint events and probabilities. The event $latex A \cap B$ is called a joint event and its probability $latex P(A \cap B)$ is called a joint probability.

Comments. (1) Following the recommendation made earlier, remember that $latex A$ and $latex B$ may vary and $latex P(A \cap B)$ is a function of two sets.

(2) In the wheat example we have four possible states of nature (the basic events) $latex H \cap S$, $latex H \cap \bar S$, $latex \bar H \cap S$, $latex \bar H \cap \bar S$ and, correspondingly, four probabilities $latex P(H \cap S)$, $latex P(H \cap \bar S)$, $latex P(\bar H \cap S)$, $latex P(\bar H \cap \bar S)$. By the completeness axiom, these probabilities sum to 1. As usual, this information can be displayed in a table.

Table 4.5. One-dimensional table of joint events and probabilities.

By the complement rule $latex P(S) + P(\bar S) = 1,\ P(H) + P(\bar H) = 1.$ Therefore we have two more tables, see Table 4.6.

Table 4.6. Separate probability tables.

(3) It is better to replace a 1-D Table 4.5 by a 2-D Table 4.7, in order to see better its link to Tables 4.6.

Table 4.7. Joint probability table.

Own and marginal probabilities. It is convenient to call the numbers $latex P(S)$ and $latex P(\bar S)$ own probabilities of the pair $latex {S,\,\bar S}$ and the numbers $latex P(H)$ and $latex P(\bar H)$ own probabilities of the pair $latex {H,\,\bar H}$. The customary name for own probabilities, marginal probabilities, reflects their placement in the table but not their role.

In a 2-D table it is easier to form pairs of individual events $latex S$, $latex \bar S$, $latex H$ and $latex \bar H$. The probabilities from Table 4.5 have migrated to the central part of Table 4.7. By the assembly formula

$latex P(S \cap H) + P(S \cap \bar H) = P(S),\ P(\bar S \cap H) + P(\bar S \cap \bar H) = P(\bar S).$

Thus, row sums of the probabilities in the central part of Table 4.7 are equal to own probabilities of the pair $latex {S,\bar S}$ (displayed in the right margin). Similarly, column sums of the probabilities in the central part of Table 4.7 are equal to own probabilities of the pair $latex {H,\bar H}$ (displayed in the lower margin).

(4) The unity in the lower right corner of Table 4.7 is

• (a) the sum of joint probabilities in the central part,
• (b) the sum of own probabilities in the right margin and
• (c) the sum of own probabilities in the lower margin.

(5) A cross-table (also called a contingency table) is similar to the joint probabilities table, except that absolute frequencies are used instead of probabilities.

Exercise 4.6.

(a) List and prove 5 identities which hold for a joint probabilities table.

(b) Draw a joint probabilities table for the case when in the first column there are events $latex {A_1}$, ..., $latex {A_n}$ and in the first row there are events $latex {B_1}$, ..., $latex {B_m}$ ($latex n$ is not necessarily equal to $latex m$).

Summation signs. Because of importance of the last exercise, we'll do it together. This is the place to learn the techniques of working with summation signs. The sum

$latex {a_1} + ... + {a_n} \qquad (\textrm{Eq. 4.3})$

(in which the three points mean "and so on") is written as

$latex \sum\limits_{i = 1}^n {{a_i}}. \qquad (\textrm{Eq. 4.4})$

Eq. (4.3) can be referred to as an extended expression and Eq. (4.4) is a short expression. In the short expression we indicate the index that changes, $latex i$, and that it runs from the lower limit of summation, 1, to the upper limit of summation, $latex n$. If you have problems working with summation signs, replace them with extended expressions, do the algebra, and then convert everything back. For the beginning remember two things:

1. The sum Eq. (4.4) does not depend on $latex i$, as one can see from the extended expression. In particular, the summation index can be replaced by anything else, for example, $latex \sum\nolimits_{j = 1}^n {{a_j}}$ is the same thing as Eq. (4.4).
2. When more than one different indices are involved, it is better to use different notations for them.

The answer to Exercise 4.6(b) is given in Table 4.8. Each of the families $latex {{A_1},\,...,\,{A_n}}$ and $latex {{B_1},\,...,\,{B_m}}$ in that table is mutually exclusive and collectively exhaustive.

Table 4.8. Joint probability table (general case).

Identities:

1. Sums across rows: $latex \sum\nolimits_{j = 1}^m P({A_i} \cap {B_j}) = P({A_i})$, $latex i=1,...,n$.
2. Sums across columns: $latex \sum\nolimits_{i = 1}^n P({A_i} \cap {B_j}) = P({B_j}),\ j = 1,...,m.$
3. Sums of own probabilities: $latex \sum\nolimits_{j = 1}^m P({B_j}) = 1,\ \sum\nolimits_{i = 1}^n P({A_i}) = 1$.
4. Sum of joint probabilities: $latex \sum\nolimits_{j = 1}^m \sum\nolimits_{i = 1}^n P({A_i} \cap {B_j}) = 1.$

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