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Topics: Probability/Equally Likely

Prerequisites:

This activity explores the meaning of probability and equally likely outcomes again through physical and computer simulations. It does not assume much familiarity with the software, Minitab. Note: Students will be shown how to create a Minitab macro in this lab.

Materials:

Four index cards and one sheet of scratch paper per student

Goals:

• To develop intuition for probability as long-term relative frequency

• To learn some elementary counting rules

• To emphasize consideration of when the "equal likeliness" assumption applies and when it does not

• To gain an introduction to the concepts of sample space, random variable, and expected value

Situation:

Suppose that on one night at a certain hospital, four mothers (named Johnson, Miller, Smith, and Williams) give birth to baby boys. Each mother gives her child a first name alliterative to his last: Jerry Johnson, Marvin Miller, Sam Smith, and Willy Williams. As a very sick joke, the hospital staff decides to return babies to their mothers completely at random. We will first use simulation to investigate what will happen in the long run.

Simulation Analysis:

(a) Take four index cards and one sheet of scratch paper. Write a baby’s first name on each index card, and divide the sheet of paper into four areas with a mother’s last name written in each area. Shuffle the four index cards well, and then deal them out randomly with one index card going to each area of the sheet. Finally, turn over the cards to reveal which babies were randomly assigned to which mothers. Record how many mothers got the right baby.

# of matches:

(b) Do the random “dealing” of babies a total of five times, recording in each case the number of matches:

Repetition # 1 2 3 4 5

# of matches

(c) Combine your results on the number of matches with the rest of the class, obtaining a tally of how often each result occurred. Record the counts and proportions in the table below:

# of matches 0 1 2 3 4 Total

Count

Proportion 1.00

(d) In what proportion of these simulated cases did at least one mother get the correct baby?

The probability of a random event is the long-run proportion (or relative frequency) of times the event would occur if the random process were repeated over and over under identical conditions. One can approximate a probability by simulating the process a large number of times. Simulation leads to an empirical estimate of the probability.

(Exact) Enumeration Analysis:

In situations where the outcomes of a random process are equally likely to occur (e.g. tossing a fair coin), exact probabilities can be calculated by listing all of the possible outcomes and determining the proportion of these outcomes which correspond to the event of interest. The listing of all possible outcomes is called the sample space.

The sample space for the “random babies” consists of all possible ways to distribute the four babies to the four mothers. Let xyzw mean that the baby x went to the first mother, baby y to the second mother, baby z to the third mother, and baby w to the fourth mother. For example, 1243 would mean that the first two mothers got the right baby, but the third and fourth mothers had their babies switched.

(e) Below is the beginning of a list of the sample size for the “random babies” process. Fill in the remaining possibilities, using this same notation. [Try to be systematic about how you list these outcomes so that you don’t miss any. One sensible approach is to list in a second row the outcomes for which mother 1 gets baby 2 and then in the third row the cases where mother 1 gets baby 3 and so on.]

Sample Space:

1234 1243 1324 1342 1423 1432

(f) How many possible outcomes are there in this sample space? That is, in how many different ways can the four babies be returned to their mothers?

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. 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. Since 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