Understanding independence and conditional probability
• Understand that two events A and B are independent if P(A and B) = P(A) • P(B) (S-CP.A.2).
• Understand the conditional probability of A given B as P(A and B)/P(B) (S-CP.A.3).
• Determine whether pairs of events are independent (S-CP.A.2).
• Find the conditional probability of A given B as the fraction of B’s outcomes that also belong in A (S-CP.B.6).
• Recognize independence in everyday situations and explain it in everyday language (S-CP.A.5).
In this section, uniform probability models are the main context. Students examine situations involving pairs of independent and non-independent events, allowing them to learn to distinguish between such pairs. They calculate probabilities of compound events in uniform probability models, finding the conditional probability of A given B as the fraction of B’s outcomes that are also in A. They note relationships such as P(A|B) = P(A and B)/P(B), and, when A and B are independent, P(A) = P(A|B).
WHAT: Students are presented with a scenario about two different kinds of envelopes (four red and four blue) in which a \$1 bill is placed in four of the envelopes. They calculate probabilities of events such as “you pick a lucky envelope given that you have picked a blue envelope” under various conditions, e.g., one \$1 bill is placed in a blue envelope and the rest in red envelopes or two bills are placed in blue envelopes and two in red envelopes. These are conditional probabilities but because students know the sample space, they have no need to use the formula P(A|B) = P(A and B)/P(B).
This task introduces the definition of independence as: “Two events are independent if knowing that one event has occurred has no effect on the probability that the other has occurred.” They then are asked to consider the implication of events E and F being independent on the values of P(E) and P(E|F).
WHY: Placed here, this task introduces a definition of independence and its implications for conditional probability, namely the implication that P(E) and P(E|F) are equal if E and F are independent. Because the task involves events of the form “A given B,” it is an opportunity to introduce the term “conditional probability” and associated notation E|F. It is also an opportunity to notice that the conditional probability of A given B is the fraction of B’s outcomes that also belong in A. It does not spell out the implication that P(A and B) = P(A) • P(B), if A and B are independent.
WHAT: Students are asked to calculate probabilities of outcomes of two pairs of events in a context that is readily associated with a sample space and uniform probability model. They are then are asked if the pairs are independent or not. One pair is independent, and one is not.
WHY: Students work with the understanding of independence gained from the previous task, that if E and F are independent, then P(E) and P(E|F) are equal (the second approach listed in the task commentary).
WHAT: Students are told “On school days, Janelle sometimes eats breakfast and sometimes does not. After studying probability for a few days, Janelle says, "The events 'I eat breakfast' and 'I am late for school' are independent.” Students are asked to explain what this means in everyday language.
WHY: Students apply their understanding of independence in a context that is more abstract than those of the previous tasks because it does not have a readily associated uniform probability model (as in the tasks "Lucky Envelopes" or "Cards and Independence") (MP.2)