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.

Women / Men
Person category / Number aboard / Number saved / Number lost / Person category / Number aboard / Number saved / Number lost
1st Class / 144 / 140 / 4 / 1st Class / 175 / 57 / 118
2nd Class / 93 / 80 / 13 / 2nd Class / 168 / 14 / 154
3rd Class / 165 / 76 / 89 / 3rd Class / 462 / 75 / 387
Crew / 23 / 20 / 3 / Crew / 885 / 192 / 693
Total / 425 / 316 / 109 / Total / 1690 / 338 / 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.)
  1. Why should you exclude the crew member data from your analysis about the influence of passenger class of travel on survival rate?
  1. Compute some conditional probabilities using your table in #1.
  1. P(Survived|Passenger) = ______
  2. P(3rdClass|Passenger) = ______
  3. P(Survived|2ndClass) = ______
  4. P(3rdClass|Died) = ______
  5. P(1stClass|Survived) = ______
  1. 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.

  1. P(Survived|Passenger) =
  1. P(3rdClass|Passenger) =
  1. P(Survived|2ndClass) =
  1. P(3rdClass|Died) =
  1. P(1stClass|Survived) =
  1. Was the survival rate of the passengersindependent from their class of travel? Give evidence to show that class of travel was or was not independent from survival rate.
  1. 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 student-generated table (but not the original one).

Here are standard responses:

Counts / Survived / Died / TOTAL
1st Class / 197 / 122 / 319
2nd Class / 94 / 167 / 261
3rd Class / 151 / 476 / 627
TOTAL / 442 / 765 / 1207
Fractional
Probabilities / Survived / Died / TOTAL
1st Class / 197/1207 / 122/1207 / 319/1207
2nd Class / 94/1207 / 167/1207 / 261/1207
3rd Class / 151/1207 / 476/1027 / 627/1207
TOTAL / 442/1207 / 765/1207 / 1207/1207
Percentage
Probabilities / Survived / Died / TOTAL
1st Class / 16.3% / 10.1% / 26.4%
2nd Class / 7.8% / 13.9% / 21.7%
3rd Class / 12.5% / 39.4% / 51.9%
TOTAL / 36.6% / 63.4% / 100%
Decimal Probabilities / Survived / Died / TOTAL
1st Class / 0.163 / 0.101 / 0.264
2nd Class / 0.078 / 0.139 / 0.217
3rd Class / 0.125 / 0.394 / 0.519
TOTAL / 0.366 / 0.634 / 1.000
  1. P(Survived|Passenger) = P(Survived and Passenger)/P(Passenger) = 0.366/1 = 0.366
  2. P(3rdClass|Passenger) = P(3rd Class and Passenger)/P(Passenger) = 0.519/1 = 0.519
  3. P(Survived|2ndClass) = P(Survived and 2ndClass)/P(2ndClass) = 0.078/0.217 = 0.36
  4. P(3rdClass|Died) = P(3rdClass and Died)/P(Died) = 0.394/0.634 = 0.62
  5. P(1stClass|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 / Survived / Died / TOTAL
1st Class / 0.097 / 0.167 / 0.264
2nd Class / 0.079 / 0.138 / 0.217
3rd Class / 0.190 / 0.329 / 0.519
TOTAL / 0.366 / 0.634 / 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.