Probability - the Australian Mathematical Sciences Institute

{24} • ProbabilitySuppose the chance that I leave my keys at home and arrive at work without them is 0.01.Now suppose I was up until 2 a.m. the night before, had to hurry in the morning, and useda different backpack from my usual one. What then is the chance that I don’t have mykeys? What we are given — new information about what has occurred — can change theprobability of the event of interest. Intuitively, the more difficult circumstances of themorning described will increase the probability of forgetting my keys.Consider the probability of obtaining 2 when rolling a fair die. We know that the probabilityof this outcome is 1 6. Suppose we are told that an even number has been obtained.Given this information, what is the probability of obtaining 2? There are three possibleeven-number outcomes: 2, 4 or 6. These are equiprobable. One of these three outcomesis 2. So intuition tells us that the probability of obtaining 2, given that an even numberhas been obtained, is 1 3. Similar reasoning leads to the following: The probabilityof obtaining 2, given that an odd number has been obtained, equals zero. If we knowthat an odd number has been obtained, then obtaining 2 is impossible, and hence hasconditional probability equal to zero.Using these examples as background, we now consider conditional probability formally.When we have an event A in a random process with event space E , we have used thenotation Pr(A) for the probability of A. As the examples above show, we may need tochange the probability of A if we are given new information that some other event D hasoccurred. Sometimes the conditional probability may be larger, sometimes it may besmaller, or the probability may be unchanged: it depends on the relationship betweenthe events A and D.We use the notation Pr(A|D) to denote ‘the probability of A given D’. We seek a way offinding this conditional probability.It is worth making the observation that, in a sense, all probabilities are conditional: thecondition being (at least) that the random process has occurred. That is, it is reasonableto think of Pr(A) as Pr(A|E ).It helps to think about conditional probability using a suitable diagram, as follows.û ûûû û û ûû ûûû ûû ûûûû û û ûûû ûû ûû û û ûA ûûû û û ûû û û û Dû û ûεIllustrating conditional probability.