2 - TWO OR MORE RANDOM VARIABLES Many problems in probability theory relate to two or more random variables. You might have equipment which is controlled by two computers and you are interested in knowing the probability of both computers failing at once. An important consideration is whether or not the computers fail independently. A hardware failure in one computer is unlikely to provoke a failure in the other but a power cut might cause the simultaneous failure of both computers. Let's start by considering two dice. . . Two Random Variables When two dice are thrown, you can either think about a single outcome which is one of the 36 possibilities 1,1 to 6,6 or you can think of a separate outcome for each die. It is more usual to take the second view and, in any analysis, use two random variables, each of which can take one of the 6 possibilities 1 to 6. The Multiplication Theorem of Probability What is the probability of throwing two 6s? If both dice are fair, the answer is 1. This result is an application of the multiplication theorem of probability which, in the present case, says that the overall probability is the product of the individual probabilities. The probability of the first die showing 6 is 1 6 and the probability of the second die showing 6 is also 1 and the product 1 1 1 6 36 . Independence and the Multiplication Theorem - I The multiplication theorem requires the two throwings to be independent. Informally this means that the outcome from one die has no influence on the outcome of the other. It seems most unlikely that one die could affect the other but it is easy to imagine a closely analogous case where independence couldn't be so readily assured. Consider a two-drum fruit machine where, instead of sporting fruit icons, each drum has six segments numbered 1 to 6. If the machine were well set up the two drums would be independent and behave like two dice but if grit gets into the works independence may be compromised. In an extreme case the two drums might stick together in such a way that they always showed the same number; the probability of obtaining two 6s now would simply be z. Independence - Formal Definition Two events A and B are said to be independent if and only if: P(B A) = P(B) In simple terms, knowing that event A has occurred makes it neither more likely nor less likely that event B has occurred. In particular, knowing that the first die shows 6 makes it neither more nor less likely that the second die shows 6. - 2.1 -
Independence can be an important consideration even when there is just a single random variable. Suppose a card is selected from a pack of cards. The probability of the event 'an ace has been drawn' is - and the probability that 'an ace has been drawn given that a