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ProbabilityPHYS 1301 F99Prof. T.E. Coanversion: 15 Sep ’98The naked hulk alongside came,And the twain were casting dice;“The game is done! I’ve won! I’ve won!”Quoth she, and whistles thrice.Samuel Taylor Coleridge, Rime of the Ancient MarinerMost of us have had the experience of listening to the weather report andhearing at one time or another the announcer say ”the chance of rain tomorrow is70%.” What does this statement mean? Intuitively, you might say that it is more likelythan not that it will rain tomorrow. If we somehow managed to experience many dayslike today, then we would expect that more often than not it would rain the next day.This weather forecast, like all statements about chance, is a kind of guess. Ourignorance prevents us from making a firm statement about whether or not it definitelywill rain the next day. The theory of probability permits us to make sensible andquantitative guesses about matters that have a consistent average behavior.To be clear about what the word “probability” really means and how youactually calculate it, consider the case where you throw two dice, one green and onered. Each die face can show any integer in the range from 1 to 6 and you areinterested in the sum of the two die faces. You want to know what the “probability” ofrolling a 5 is because if you can reliably determine this probability you will win muchmoney and the love of your dreams. To begin with, what happens when you roll thepair of dice? Well, the outcome – the sum of the die face - can be any integer between2 and 12. In our example, there are a number of different ways that a 5 can be rolled.Call this set of different ways A and we have A = {(1,4), (2,3), (3,2), (4,1)},where the first number in each ordered pair is the number the red die shows and thesecond number is the number the green die shows. Each ordered pair of numbers iscalled an outcome and each roll of the die pair is called an “experiment.” You shouldbe able to convince yourself that there are a total of 36 possible outcomes whenrolling the dice. The red die can show any integer from 1 to 6 and since for each ofthese numbers the green die can show any number from 1 to 6, 6 X 6 makes 36.Useful jargon is that the set of all possible outcomes in an experiment is called the“sample space.” For our experiment of rolling two dice at a time, the sample space isthe set of 36 possible outcomes or ordered pairs of numbers. Do not confuse thesample space with the total number of times you happen to roll the dice, which couldbe 65 times or 500,000 times.What does this have to do with calculating the probability of rolling a 5? Well,by “probability” of a particular outcome of an experiment, we mean our estimate ofthe most likely fraction of a number of repeated observations that will yield thatparticular outcome. And do how do you calculate this probability? If you think thateach outcome is equally likely, you simply sum up the number of outcomes that willyield a particular event and then divide by the size of the sample space. In ourexample, there are 4 possible outcomes that produce the “event” of rolling a 5 andthere are 36 total possible outcomes in our sample space, so the probability of rollinga 5 is 4/36 = 1/9. Symbolically, we can write that the probability of event A,P A =NA/N. Here,NA is the number of outcomes that produce the event A andN is the size of the sample space. There are subtleties you should be aware of. To assign a probability to someoutcome, it is necessary that the experiment be capable of being repeated. Forexample, it is far from clear that the statement, “the probability that David Grahammurdered Adrianne Jones is 65%” (a local murder trial in progress), has any meaningat all. How do we arrange to run many “experiments” with the participation of thedefendant and the deceased? Is the deceased supposed to be repeatedly resurrectedafter her murder so that the experiments can continue? Secondly, as more informationbecomes available to us, our probability estimate for a particular outcome in theexperiment can change. Suppose the experiment is that your sister draws a card froma standard deck and then asks you the probability that it is a queen. If you find outsomehow that your sister nervously twitches her ears when she draws either aces orqueens, then your answer will certainly depend on the motion of her ears. Having theextra information doesn’t change the experiment in any way (your sister twitches herears whether you know it or not), it does however change your knowledge of theexperiment.When we are playing with our dice, we do not necessarily expect that if weroll the dice 45 times we will observe that exactly 1/9 of the time the sum of the diefaces will be 5, even if the dice are honest. This does not mean that our notions ofprobability are useless. It does mean that to make a probabilistic statement impliesthat we have a certain amount of ignorance of the experimental situation. If wesomehow knew more, we could say exactly what the dice were going to do. However,we can say that if we keep rolling the dice, we do expect that the fraction of times thedie face sum to 5 will indeed approach 1/9. So, how many times do we have to roll the dice before we are confident thatour probability calculation is really correct? The answer is there is no specific numberof times we need to roll the dice that will definitely tell us one way or the other thatour probability calculation is absolutely correct! The reason is that there is somechance, no matter how small, that the dice after many throws happen to come upsumming to 5 at a rate different from 1/9. (For example, if you threw the dice 99,000times, it is certainly possible that the number of times the dice summed to 5 could bedifferent from 11,000 even if there is no cheating.) The important point here is that weexpect that the more often we roll the dice, the more likely the summed resultsapproach our probabilistic predictions. Differences between the actual results of ourexperiments and our probabilistic predictions are called “ statistical fluctuations.” Ifour probabilistic predictions are sensible, then we expect the statistical fluctuations tobecome smaller as the number of times we perform the experiment becomes larger,i.e., the larger the “statistics” we collect.To summarize, in today’s lab we want to check two important ideas aboutprobability. First, we want to check this idea that


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SMU PHYS 1301 - Probability

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