Page 41 - Probability and Statistical Inference

18 1. Nations of Probability

occupy the second and third or the third and fourth or the fourth and fifth
seats while the other three friends would take the remaining three seats in any
order. One can also permute John and Molly in 2! ways. That is, we have
P(John and Molly sit next to each other) = (2!)(4)(3!)/5! = 2/5.!
Example 1.4.12 Consider distributing a standard pack of 52 cards to four
players (North, South, East and West) in a bridge game where each player gets
13 cards. Here again the ordering of the cards is not crucial to the game. The
total number of ways these 52 cards can be distributed among the four players
is then given by . Then, P(North will receive 4 aces and 4
kings) = = ,
because North is given 4 aces and 4 kings plus 5 other cards from the remaining
44 cards while the remaining 39 cards are distributed equally among South,
East and West. By the same token, P(South will receive exactly one ace) =
= , since South would re-
ceive one out of the 4 aces and 12 other cards from 48 non-ace cards while the
remaining 39 cards are distributed equally among North, East and West. !

1.5 Discrete Random Variables

A discrete random variable, commonly denoted by X, Y and so on, takes a
finite or countably infinite number of possible values with specific probabili-
ties associated with each value. In a collection agency, for example, the man-
ager may look at the pattern of the number (X) of delinquent accounts. In a
packaging plant, for example, one may be interested in studying the pattern of
the number (X) of the defective packages. In a truckload of shipment, for ex-
ample, the receiving agent may want to study the pattern of the number (X) of
rotten oranges or the number (X) of times the shipment arrives late. One may
ask: how many times (X) one must drill in a oil field in order to hit oil? These
are some typical examples of discrete random variables.
In order to illustrate further, let us go back to the Example 1.1.2. Sup-
pose that X is the total score from the red and yellow dice. The possible
values of X would be any number from the set consisting of 2, 3, ..., 11, 12.
We had already discussed earlier how one could proceed to evaluate P(X =
x), x = 2, ..., 12. In this case, the event [X = 2] corresponds to the subset
{11}, the event [X = 3] corresponds to the subset {12, 21}, the event [X =
4] corresponds to the subset {13, 22, 31}, and so on. Thus, P(X = 2) = 1/36,
P(X = 3) = 2/36, P(X = 4) = 3/36, and so on. The reader should