Titanic Tragedy-Part III:
A Solidify Understanding Task
Class of travel may also have influenced the survival rate, where first-class passengers received
special treatment in boarding the lifeboats, while some other passengers were prevented from
boarding because of lack of space.
Person
category
1st Class
2nd Class
3rd Class
Crew
Total
Women
Number
Number
aboard
saved
144
140
93
80
165
76
23
20
425
316
Number
lost
4
13
89
3
109
Person
category
1st Class
2nd Class
3rd Class
Crew
Total
Men
Number Number
aboard
saved
175
57
168
14
462
75
885
192
1690
338
Number
lost
118
154
387
693
1352
1. Put the data above into a two-table that compares the variables of Class and Survival. (Do
not include the crew member data in your analysis.)
2. Why should you exclude the crew member data from your analysis about the influence of
passenger class of travel on survival rate?
3. Compute some conditional probabilities using your table in #1.
a. P(Survived|Passenger) = _______________________________________
b. P(3rdClass|Passenger) = _________________________________________
c. P(Survived|2ndClass) = _________________________________________
d. P(3rdClass|Died) = ________________________________________
e. P(1st Class|Survived) = _______________________________________
4. The conditional probability of two events A and B can also be calculated as follows:
P(A|B) = P(A and B)/P(B)
Use this relationship to check your answers to #3 above. Clearly show how you used the
boldface equation in each case.
a. P(Survived|Passenger) =
b. P(3rdClass|Passenger) =
c. P(Survived|2ndClass) =
d. P(3rdClass|Died) =
e. P(1st Class|Survived) =
5. Was the survival rate of the passengers independent from their class of travel? Give
evidence to show that class of travel was or was not independent from survival rate.
6. The independence of two variables A and B means that their joint probability (P(A and B))
is equal to the product of their marginal probabilities—P(A)P(B). Use this idea to create a
two-way table that uses the same marginal probabilities as #1, but which clearly
demonstrates that class and survival are independent of one another.
Titanic Tragedy-Part III: – Teacher Notes
A Solidify Understanding Task
Purpose: Students will practice their understanding of how to test for independence between
variables, given data in a two-way table. They will apply it in a situation larger than a 2 by 2 table.
Core Standards Focus:
S.CP.2: Understand that two events A and B are independent if the probability of A and B occurring
together is the product of their probabilities, and use this characterization to determine if they are
independent.
S.CP.3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret
independence of A and B as saying that the conditional probability of A given B is the same as the
probability of A, and the conditional probability of B given A is the same as the probability of B.
S.CP.4: Construct and interpret two-way frequency tables of data when two categories are
associated with each object being classified. Use the two-way table as a sample space to decide if
events are independent and to approximate conditional probabilities. For example, collect data from
a random sample of students in your school on their favorite subject among math, science, and English.
Estimate the probability that a randomly selected student from your school will favor science given
that the student is in tenth grade. Do the same for other subjects; compare the results.
Related Standards: S.CP.5
Launch (Whole Class):
Prompt the whole class: “Do you think that survival on the Titanic only depended on gender, or
could there have been other variables upon which survival depended?” Allow a minute of
consultation with a partner, then take ideas in popcorn fashion from the partnerships until all ideas
have been collected. You should hear such things as “being a member of the crew or not” and “the
class of the passenger” and perhaps other variables that do not figure in our data.
Explore (Small Group):
Distribute Titanic Tragedy Part III. As you monitor student work on the task, look for different choices
made by groups for their two-way tables in #1. They might use 1) counts, 2) fractions, or 3) percentages
in the cells of the table (although they should not mix the forms). As an informal assessment see if
students can describe in words the meanings of particular cells in their tables (e.g. “the number of 1st class
passengers that died” or “the fraction of passengers that were 3rd class).
Insist that students write down how they computed the conditional probabilities in #3. Note that,
although passengers are only a portion of the original data, they should comprise the entirety of the data
table that students constructed in #1, so that P(Survived|Passenger) = P(Survived) for the studentgenerated table (but not the original one).
Here are standard responses:
Counts
1st Class
2nd Class
3rd Class
TOTAL
Survived
197
94
151
442
Died
122
167
476
765
TOTAL
319
261
627
1207
Fractional
Probabilities
1st Class
2nd Class
3rd Class
TOTAL
Survived
Died
TOTAL
197/1207
94/1207
151/1207
442/1207
122/1207
167/1207
476/1027
765/1207
319/1207
261/1207
627/1207
1207/1207
Percentage
Probabilities
1st Class
2nd Class
3rd Class
TOTAL
Survived
Died
TOTAL
16.3%
7.8%
12.5%
36.6%
10.1%
13.9%
39.4%
63.4%
26.4%
21.7%
51.9%
100%
Decimal
Probabilities
1st Class
2nd Class
3rd Class
TOTAL
Survived
Died
TOTAL
0.163
0.078
0.125
0.366
0.101
0.139
0.394
0.634
0.264
0.217
0.519
1.000
a. P(Survived|Passenger) = P(Survived and Passenger)/P(Passenger) = 0.366/1 = 0.366
b. P(3rdClass|Passenger) = P(3rd Class and Passenger)/P(Passenger) = 0.519/1 = 0.519
c. P(Survived|2ndClass) = P(Survived and 2ndClass)/P(2ndClass) = 0.078/0.217 = 0.36
d. P(3rdClass|Died) = P(3rdClass and Died)/P(Died) = 0.394/0.634 = 0.62
e. P(1st Class|Survived) = P(1stClass and Survived)/P(Survived) = 0.163/0.366 = 0.45
Also, select and sequence student work on the question of independence of class and survival. You might
sequence the groups from most informal arguments to most formal mathematical arguments or based on
the mode of representation used.
Discuss (Whole Class):
Allow students from groups using different choices in their tables to display their tables for
comparison. Encourage the class to specify what are the advantages of each choice. The class may
come up with a variety of ideas here, but be sure that the idea that the probability version makes it
quicker to assess independence comes out.
Have a pair of students come to the board and write out their work on 3c and 4c, and lead a short
discussion of the two methods that arrived at the same answer.
Bring up students to present their work on the independence of class and survival in the sequence
you’ve selected (#5 and #6). Make sure that two ideas are highlighted:
1) The idea that independence requires that the conditional probabilities equal, more or less,
the unconditional probabilities—that is, that P(A | B) = P(A) and P(B|A) = P(B). This will be
evident in most groups work on #5.
2) The idea that independence requires that the joint probability equals, more or less, the
product of the marginal probabilities—that is, P(A and B) = P(A)P(B) when A and B are
independent. If it hasn’t been surfaced in discussing #5, this is the explicit target of #6.
Here’s the correct version for #6:
Probabilities
1st Class
2nd Class
3rd Class
TOTAL
Survived
0.097
0.079
0.190
0.366
Died
0.167
0.138
0.329
0.634
TOTAL
0.264
0.217
0.519
1.000
Note that in Secondary III students will build statistical inference upon a comparison of
actual data (such as table in #1) with an idealized set of data as if the variables were
perfectly independent (such as table in #6). They will ask the question of how different
does the actual data have to be from the ideal in order to infer that the variables show a
dependence upon one another.