Probability - the Australian Mathematical Sciences Institute

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{12} • ProbabilityExercise 1Consider rolling two normal dice and observing the uppermost faces obtained.aFor this random procedure, which of the following are events?iiiiiiivvA = “the sum of the two numbers observed is a prime number”B = “Barry sneezes just before the dice are rolled”C = “at least one of the two dice shows an odd number”D = “at least one of the two dice shows a six”E = “the outcome (6,6) is observed”b Which outcomes are in the event A?c How many outcomes are in D?dDescribe A ∩ E in words.Exercise 2Observations are made by the Bureau of Meteorology on a city’s weather every day, in asystematic fashion. Suppose that we define a random procedure to be: record the dailyamounts of rainfall, to the nearest tenth of a millimetre, in August 2017.abFor this random procedure, which of the following are events?i A = “the only days with recorded rainfall greater than 0.0 mm are Mondays”ii B = “the main street floods on Tuesday 15 August”iii C = “the total rainfall for the month is greater than 20.0 mm”Consider the following events:• D = “the rainfall amount recorded on 31 August is greater than the total recordedfor the first 30 days”• E = “no rainfall is recorded before the last seven days of the month”• F = “the rainfall amount on Wednesday 16 August is 18.7 mm”.Which of the following pairs of events are mutually exclusive?i A and E ii A and F iii D and E iv E and FcDescribe the following events in words:i A ′ ii A ∩ E iii A ∪C .
A guide for teachers – Years 11 and 12 • {13}The probability axiomsWe now consider probabilities of events. What values can a probability take, and whatrules govern probabilities?Some notation: We have used capital letters A,B,... for events, and we have observedthat probability can be thought of as a function that has an event as its argument. Thedomain is the collection of all events, that is, all possible subsets of E . By analogy withusual function notation, such as f (x), we use the notation Pr(A) to denoted the probabilityof the event A.There are just three basic rules or axioms for probabilities, from which several other importantrules can be derived. These fundamental rules were first spelled out in a formalway by the Russian mathematician Andrey Kolmogorov.The three axioms of probability1 Pr(A) ≥ 0, for each event A.2 Pr(E ) = 1.3 If events A and B are mutually exclusive, that is, if A ∩ B = ∅, thenPr(A ∪ B) = Pr(A) + Pr(B).The first axiom fits with our experience of measuring things, like length and area. Just asthe lowest possible value for a length is zero, the lowest possible value for a probability iszero. Any other choice for the minimum numerical value of a probability would not work.The second axiom says that the probability that something will happen is one. Anotherway of thinking about the second axiom is this: When the random process is carried out,we are certain that one of the outcomes in the event space E will occur.The third axiom determines the way we work out probabilities of mutually exclusiveevents. The axiom says that, if A and B are mutually exclusive, then the probability that atleast one of them occurs is the sum of the two individual probabilities. While this seemsvery compelling, it cannot be proved; mathematically, it must be assumed.Taken together, the three axioms imply that probabilities must be between zero and one(inclusive): 0 ≤ Pr(A) ≤ 1, for any event A. We will prove this in the next section.The choice of this numerical range for probabilities fits with relative frequencies fromdata, which are always in this range. In earlier years, students learned how relative frequenciesare estimates of probabilities. We may estimate the probability of being lefthandedby finding the relative frequency, or proportion, of left-handed people in a randomsample of people. Any relative frequency of this sort must be a fraction betweenzero and one; it cannot be negative, and it cannot be greater than one. This basic observationfits with the range for probabilities themselves: 0 ≤ Pr(A) ≤ 1.
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Probability - the Australian Mathematical Sciences Institute

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