invest_3ed.pdf

# First lets list out all possible outcomes for

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First, let’s list out all possible outcomes for returning four babies to their mothers at random. We can organize our work by letting 1234 represent the outcome where the first baby went to the first mother, the second baby to the second mother, the third baby to the third mother, and the fourth baby to the fourth mother. In this scenario, all four mothers get the correct baby. As another example, 1243 means that the first two mothers get the right baby, but the third and fourth mothers have their babies switched. Definition: A sample space is a list of all possible outcomes of a random process. All of the possible outcomes are listed below:

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Chance/Rossman, 2015 ISCAM III Investigation B 15 Sample Space: 1234 1243 1324 1342 1423 1432 2134 2143 2314 2341 2413 2431 3124 3142 3214 3241 3412 3421 4123 4132 4213 4231 4312 4321 In this case, returning the babies to the mothers completely at random implies that the outcomes in our sample space are equally likely to occur ( outcome probability = 1 / number of possible outcomes ). (o) How many different outcomes are there for returning four babies to their mothers? What is each outcome ’s probability of occurring for any trial? You could have determined the number of possible outcomes without having to list them first. For the first mother to receive a baby, she could receive any one of the four babies. Then there are three babies to choose from in giving a baby to the second mother. The third mother receives one of the two remaining babies and then the last baby goes to the fourth mother. Because the number of possibilities at one stage of this process does not depend on the outcome (which baby) of earlier stages, the total number of possibilities is the product 1 2 3 4 = 24. This is also known as ! 4 , read “4 factorial.” Because the above outcomes are equally likely, the probability of any one of the above outcomes occurring is 1/24. Although these 24 outcomes are equally likely, we were more interested above in the probability of 0 matches, 1 match, etc. Definition: A random variable maps each possible outcome of the random process (the sample space) to a numerical value. We can then talk about the probability distribution of the random variable. These random variables are usually denoted by capital roman letters, e.g., X , Y . A random variable is discrete if you can list each individual value that can be observed for the random variable. (p) For each of the above outcomes, indicate (next to the outcome above) how many mothers get the correct baby. [ Hint : For outcome 1243, mothers one and two receive their own babies so the value of the number of correct matches is 2.] (q) In how many of the outcomes did zero mothers receive the correct baby? Probability Rule: When the outcomes in the sample space are equally likely, the probability of any one of a set of outcomes (an event) occurring is the number of outcomes in that set divided by the total number of outcomes in the sample space.
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