I must confess: I like teaching probability. I enjoy discussing concepts like fairness and expected value with students, as well as analyzing games of chance. However, even as a fan of probability, I have fallen into a few traps that seem to be pervasive in high school mathematics classrooms. One of the foremost mistakes I have made in the past is to erroneously apply the multiplication rule to find the probability of multiple events. In other words, P(A and B) = P(A) x P(B) is only valid when events is only valid when events A and B are independent, but I have applied this rule to events without attending to the condition required to use it
For some events, independence is evident. Students tend to recognize that for fair number cubes and dice, the outcome of a single toss or roll does not help us to predict the outcome of future events. Similarly, the lack of independence is sometimes evident, such as multiple draws of cards from decks without replacement. However, as students are required to compute the probability of multiple real-world events, independence or the lack thereof is often not as easy to discern, and, for various reasons, teachers may find themselves telling students to “assume independence” to facilitate neat probability calculations. By doing this, though, we may give our students the erroneous impression that the multiplication rule can be applied in all situations that do not fit the “multiple-selections without replacement” mold. To be honest, it can be difficult to assess the independence of real-world events using classic textbook probability problems, but the Eureka Math Algebra II curriculum provides excellent opportunities for the students to analyze and assess independence using two-way tables
WHAT IT LOOKS LIKE
In Module 2 of Algebra I, the students learn how to construct and interpret two-way tables. In Lesson 4 of Module 4, students work on an example introduced in an earlier lesson containing data on participation in an after-school sports program, where the data is displayed by gender. The students are challenged to determine whether being female and participating in an after-school sports program are independent events, using the data in the table below.
The students compute row-conditional probabilities from the table to compare the probability that students participate in an after-school sports program given a) the student is randomly selected from those who were surveyed, b) the randomly-selected student is a female, and c) the randomly-selected student is male.
For instance, the probability that a randomly selected student participates in an after-school sports program is:
Similarly, the probability that a randomly selected student who is female participates in an after-school sports program is:
And the probability that a randomly selected male participates in an after-school sports program is:
Given that the value calculated for the probability of a randomly-selected student participating in an after-school sports program is the same whether the gender of the student is known or not, we can say that knowing the gender of a randomly selected student does not help us predict whether he or she will participate in an after-school sports program. In other words, the events are independent. In this case, it would be appropriate to apply the multiplication rule for calculating the probability of multiple events. For instance, the probability that a randomly selected student is female AND participates in an after-school sports program is
This value corresponds to the data from the table: 232 out of 1,000 students are females who participate in an after-school sports program.
WHY THIS MATTERS
First, working through problems such as those presented in Algebra II Module 4 provide students with a deeper understanding of independence than the often stated “independence means that the outcome of the first event does not affect the outcome of the second event.” These examples demonstrate for students how they can verify independence using real-world data, which is important in situations that reach beyond spinners, coins, cards, and dice.
Second, understanding the conditions for independence may help students avoid the trap of applying the multiplication rule where it is not appropriate. The Eureka lessons provide the students with important tools to help them critically analyze probabilities reported in research, which may also tend to assume independence.
Third, examples such as those presented in Eureka Math enable the students to derive the multiplication rule based on their understanding of conditional probability. Namely, if by the definition of conditional probability
However, events A and B are independent, so knowing the outcome of event B does not help us to predict the outcome of event A. Therefore, P(A | B) = P(A). By substituting
and applying the Multiplication of Equality gives P(A) x P(B) = P(A and B)
Discovering this relationship for themselves increases the likelihood that students will understand the multiplication rule and apply it appropriately.
This post is authored by Sara Lack, a Eureka Math writer for A Story of Functions.