AP Statistics - Introduction to Probability
Random Babies:
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 the babies to their mothers completely at random.
We want to investigate questions such as, How often will at least one mother get the right baby? How
often will every mother get the right baby? What is the most likely outcome? On average, how many
mothers will get the right baby?
Since it is clearly not feasible to actually carry out this exercise over and over to investigate what
would happen in the long run, we will use simulation instead. Simulation is an artificial
representation of a random process used to study its long-term properties. We will represent the
process of distributing babies to mothers at random by shuffling and dealing cards (representing
the babies) to regions on a sheet of paper (representing the mothers).
a)
Take a sheet of paper and tear it into four equal pieces; get out another sheet of paper and leave it
whole. Write a baby’s first name on each small piece of paper, and divide the full sheet of paper
into four areas with a mother’s last name written on each area. Shuffle the small pieces of paper
well, and then deal them out randomly with one going to each area of the sheet. Finally, turn over
the cards to reveal which babies were randomly assigned to which mothers. Record the requested
information below.
Did Mrs. Johnson get the right baby? __________________
Did all the mothers get the wrong baby? _________________
How many of the mothers got the right baby? ______________
b)
Repeat the random “dealing” of babies of total of five times, recording in each case the
information requested in the chart below:
Repetition Number
1
2
3
4
5
Johnson Match?
All Wrong?
Number of Matches?
c)
Combine your yes/no results sequentially with those of your classmates until you obtain a total of
100 repetitions. For each classmate, record in the table how often in their five repetitions Johnson
received the right baby and how often all mothers received the wrong baby. Then calculate the
cumulative totals and cumulative proportions. You will transfer your information onto the
overhead slide for the class and then copy the data on the same chart on the back of this sheet for
yourself.
Pair
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
d)
Cumulative
Johnson Matches?
Repetitions Of these 5 Cum. Total Cum. Proportion
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
All Wrong?
Of these 5 Cum. Total Cum. Proportion
Plot the cumulative relative frequencies as a function of repetitions for both of these variables
below:
1.0
0.5
0.0
5
e)
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Do the relative frequencies appear to be “settling down” and approaching one particular value?
What might you guess that one value is?
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. Simulations
leads to an empirical estimate of the probability.
f)
Now combine your results on a number of matches with the rest of the class, obtaining a tally of
how often each outcome occurred. Record the counts and proportions in the table below.
Number of Matches
Count
Proportion
0
1
2
3
4
Total
1.00
g)
In what proportion of these simulated cases did at least one mother get the correct baby?
h)
Based on the simulation results of the class, what is your empirical estimate of the probability of
no matches?
i)
Based on the simulation results of the class, what is your empirical estimate of the probability of at
least one match?
j)
Explain why an outcome of exactly three matches is impossible.
k)
Is it impossible to get four matches? Would you call it rare? Unlikely? Explain.
l)
Would you consider a result of 1 matches, or of 1 match, or of 2 matches, to be unlikely? Explain.
Pair
1
Cumulative
Johnson Matches?
Repetitions Of these 5 Cum. Total Cum. Proportion
5
2
10
3
15
4
20
5
25
6
30
7
35
8
40
9
45
10
50
11
55
12
60
13
65
14
70
15
75
16
80
17
85
18
90
19
95
20
100
All Wrong?
Of these 5 Cum. Total Cum. Proportion
Part II:
In situations where the outcomes of a random process are equally likely, exact probabilities can be
calculated by listing all of the possible outcomes and counting the proportion that corresponds to
the event of interest. The listing of all possible outcomes is called the sample space.
The sample space for the “random babies” activity consists of all possible ways to distribute the four
babies to the four mothers. Let 1234 mean that the first baby went to the first mother, the second baby
went to the second mother, the third baby went to the third mother, and the fourth baby went to the fourth
mother. In this scenario all four mothers get the correct baby. As another example, 1243 would mean
that the first two mothers got the right baby, but the third and fourth mothers had their babies switched.
All of the possibilities are listed below:
1234
2134
3124
4123
a)
1243
2143
3142
4132
1324
2314
3214
4213
1342
2341
3241
4231
1423
2413
3412
4312
1432
2431
3421
4321
How many different arrangements are there for returning the four babies to their mothers?
b)
For each of these arrangements, indicate how many mothers get the correct baby:
1234: 4 matches
1243: 2 matches
1324
1342
1423
2134
2143
2314
2341
2413
3124
3142
3214
3241
3412
4123
4132
4213
4231
4312
c)
In how many arrangements is the number of “matches” equal to exactly:
4:_____________
d)
1432
2431
3421
4321
3:_____________
2:_____________
1:_____________
0:___________
Calculate the (exact) probabilities by dividing your answers to (c) by your answer to (a).
4:_____________
3:_____________
2:_____________
1:_____________
0:___________
Comment below on how closely the exact probabilities correspond to the empirical (theoretical)
probabilities from the simulation recorded in part (f) from Part I of this activity.
An Empirical Estimate from a simulation generally gets closer to the actual probability as the
number of repetitions increases. Look at the histograms below:
e)
Generally speaking, which of these three simulations produces empirical estimates closest to the
actual probabilities? Explain your answer fully.
f)
For your class simulation results summarized in (f) from Part I, calculate the average (mean)
number of matches per repetition of the process by multiplying each outcome by the number of
occurrences, summing the products, and then dividing by the total number of repetitions.
The long-run average value achieved by a numerical random process is called its expected value.
To calculate this expected value from the (exact) probability distribution, multiply each outcome by
its probability, and then add these up over all the possible outcomes.
g)
Calculate the expected number of matches from the (exact) probability distribution, and compare
that to the average number of matches from the simulated data.