Lab Activity #16 Date:

Probability

Random Books

Suppose that on one night at a certain college, four students (named Johnson, Miller, Smith, and Williams) each

left their books behind in class. Not sure who to return the books to, as they are unlabeled, the professor returns

the books to the students in a random order. We will first use simulation to investigate what will happen in the

long run if this were to happen many times.

Simulation Analysis:

1. Take four index cards and one sheet of scratch paper. Write a student’s name on each index card

representing the book they originally left behind, and divide the sheet of paper into four areas with

student’s 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 to represent how the books were returned to the

students. Finally, turn over the cards to reveal which books were randomly assigned to which students.

Record how many students got the right book.

# of matches:

2. Do the random “dealing” of books a total of five times, recording in each case the number of matches:

3. 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:

4. Below is the beginning of a list of the sample size for the “random books” 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 student #1

gets the book of student #2 and then in the third row the cases where student #1 gets the book of student

#3 and so on.]

Repetition # 1 2 3 4 5

# matches 0 0 0 0 0

# of matches 0 1 2 3 4 Total

Count

15 12 3 0 2 32

Proportion

0.469 0.375 0.094 0 0.063 1.00

STAT 1350: Elementary Statistics Names:

Lab Activity #16 Date:

Probability

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

5. How many possible outcomes are there in this sample space? That is, in how many different ways can

the four books be returned to their student owners?

4(6)= 24 possible outcomes

6. For each of the above outcomes in your sample space, indicate how many students get the correct book.

(you can write the # matches above each outcome as 0, 1, 2, 3, or 4)

1234/ 4 matches 1243/ 2 matches 1324/ 2 matches 1342/1 match 1423/1 match 1432/1 match

2134/ 2 matches 2143/ 0 matches 2314/ 1 match 2341/ 0 matches 2413/ 0 matches 2431/ 0 matches

3124/ 1 match 3142/ 0 matches 3214/ 2 matches 3241/ 1 match 3412/ 0 matches 3421/ 0 matches

4123/ 0 matches 4132/ 1 match 4213/ 1 match 4231/ 1 match 4312/ 0 matches 4321/ 0 matches

7. In how many outcomes is the number of “matches” equal to exactly:

4: 1 3: 0 2: 4 1: 9 0: 10

8. Calculate the (exact) probabilities by dividing your answers to (7) by your answer to (5). Comment on

how closely the exact probabilities correspond to the empirical estimates from the simulation above.

4 matches: 1/24= .0312

3 matches: 0/24= 0

2 matches: 4/24= .167

1 match: 9/24= .375

0 match: 10/24= .417

The “number of matches” is an example of a random variable, which is a function assigning a numerical

output to each outcome in a sample space. Here, each of the 24 outcomes has a corresponding numerical

value for “number of matches.” This is a discrete random variable in that it that can assume only a finite

STAT 1350: Elementary Statistics Names:

Lab Activity #16 Date:

Probability

number of values. The probability distribution of a discrete random variable is given by its set of

possible values and their associated probabilities.

9. Are the possible values of the “number of matches” random variable equally likely? Explain.

Yes, that’s possible if we keep repeating this sample over and over again.

10. Comment on your sample probability and the class proportion of number of matches. Which is closer

to the calculated probability distribution?

The class proportion is closer.

# of matches 0 1 2 3 4

Sample 1 0 0 0 0

Calculated

0.4583 0.3 0.167 0 .0312

Class

Proportion

0.469 0.375 0.094 0

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