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ASU MAT 142 - Properties of Probability

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Section 8.1 – Properties of Probability 333 Section 8.1 Properties of Probability A probability is a function that assigns a value between 0 and 1 to an event, describing the likelihood of that event happening. Probability retains a strong intuitive appeal: an impossible event is assigned a 0 probability of happening, and probabilities close to zero are “unlikely” events. As the probability values increase, the events are correspondingly more likely to happen, so that probabilities close to 1 are considered “likely” events, with absolutely certain events being assigned a probability of 1 (most people would say “100%”). An event that is equally likely to be true as not be true is assigned a probability of 0.5 (again, most people would say “50%”). The formal study of probability assumes a strong knowledge of sets (Preliminary Section 8.0) and their various concepts and operations. In this section it is assumed you have this knowledge of the concepts and of the terminology. Intuitive Examples and Definitions Consider the following example, whose outcome should be obvious: Example 1. A fair coin is flipped once. What is the probability it lands heads? Discussion. The “obvious” answer seems to be 50%. Note however the requirement for the coin to be fair, meaning it is not weighted or suspect in any fashion. It is true that under this condition, the probability of the coin landing heads is 50%.  The following argument supports this claim: a coin can land two possible ways: heads or tails. Assuming this is a fair coin, the two sides are (intuitively) equally likely. We want the specific outcome of “heads”, which can happen just 1 way out of the 2 possible ways. Hence, the reasonable answer of 50%. This suggests a formal way to define a probability: An experiment generates a set of (equally likely) outcomes called the sample space S. Any subset of the sample space is called an event space (or event for short), usually represented by E. Therefore, the probability that event E occurs is the number of elements in E divided by the number of elements in S: )()()(SnEnEP = where )(En represents the number of elements in E, and )(Sn represents the number of elements in S. Since )()(0 SnEn≤≤, then 1)(0≤≤EP always. In example 1 above, the argument can be made even more formally: S is the sample space (the universe) of outcomes of flipping a coin: },{ htS=, where h = heads and t = tails. The event we are interested in is “heads”, or formally, a subset of the sample space }{hE=. Since there is just one element in the event space and two elements in the sample space, it follows that the probability of the coin landing heads on one flip is ½, or 0.5, or 50%, all of which are equivalent.Section 8.1 – Properties of Probability 334 Example 2: A fair die is rolled once. What is the probability is comes up “2”? Discussion. The sample space (universe) is }6,5,4,3,2,1{=S and the event space (subset) is }2{=E. Thus, the probability of a “2” is 61)()()"2(" ==SnEnP, which agrees with intuition.  Example 3: A roulette wheel has 38 slots, numbered 1-36 and two numbered 0 and 00. What is the probability any particular slot comes up as the winner? Discussion. It’s not a requirement to list every element in the sample space when we can write 38)(=Sn and 1)(=En. Thus, the probability for any particular number coming up, including your lucky number, is 381)()()( ==SnEnEP.  Comment: The size of the sample space can become quite large, making it impractical to list every element. However, it is true that we need to know the size of the sample and event spaces exactly, and to be able to generate elements if need be, in order to properly determine probabilities. This is illustrated in the following examples and discussed further in the next section where methods of enumeration are presented to expedite these calculations. Example 4: Two fair dice are rolled once. What is the probability the sum of the two dice is 12? 7? Discussion. It is imperative to identify the correct sample space in which each element is equally likely to occur. Thus, instead of just looking at the sums, we look at every possible roll: =)6,6()6,5()6,4()6,3()6,2()6,1()5,6()5,5()5,4()5,3()5,2()5,1()4,6()4,5()4,4()4,3()4,2()4,1()3,6()3,5()3,4()3,3()3,2()3,1()2,6()2,5()2,4()2,3()2,2()2,1()1,6()1,5()1,4()1,3()1,2()1,1(S There are 36 possible outcomes. In the case of rolling a 12, there is just one way to do that, so )}6,6{(=E and 1)(=En, thus 361)()()12( ===SnEnsumP. The event space for rolling a 7 is )}1,6(),2,5(),3,4(),4,3(),5,2(),6,1{(=E. Thus, the probability of rolling a 7 is 61366)()()7( ====SnEnsumP.  Comment: You may see a short-cut to identify the size of S: there are six ways to roll the first die and six ways to roll the second die, hence there are 3666=× ways to roll two dice simultaneously. This is true and uses the Multiplication Principal (section 8.2) to determine the quantity.Section 8.1 – Properties of Probability 335 Comment: Note that “rolling two dice simultaneously” is the same as “rolling one die twice”. They produce the exact same sample space so for all intents and purposes are the exact same experiment. This helps explain why the outcomes (5,2) and (2,5) in example 4 are treated as different outcome elements. Basic Properties of Probability Many probability questions can be solved using concepts from sets. Many are quite intuitive and you may find yourself using these concepts without being aware. The next example illustrates the use of complementary sets in calculating probabilities: Example 5: Two fair dice are rolled, and the sample space is the same as that in example 4. What is the probability the sum is not 12? Is not 7? Discussion. Since there is just one way to roll a 12, there must be 35 ways to not roll a 12. Thus the probability of not rolling a 12 is 3635)()()12( ==≠′SnEnsumP. Similarly, since there are 6 ways to roll a 7, there must be 30 ways to not roll a 7, and the probability of not rolling a 7 is 653630)()()7( ===≠′SnEnsumP.  If E is an event, then E′ is the complement of E, that is, all the elements in the sample space S that are not in E. This suggests a useful property involving an event E


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ASU MAT 142 - Properties of Probability

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