Understanding independence and conditional probability
• Understand that two events A and B are independent if P(A and B) = P(A) • P(B). • Understand the conditional probability of A given B as P(A and B)/P(B). • Determine whether pairs of events are independent. • Find the conditional probability of A given B as the fraction of B’s outcomes that also belong in A. • Recognize independence in everyday situations and explain it in everyday language.
In high school the study of probability is extended to the notions of independence and conditional probability. In contrast to grade 7 where students computed P(A or B) by counting occurrences of simple events in A ∪ B, students in this unit learn to calculate P(A or B) in terms of P(A), P(B), and P(A and B). They begin by considering compound events as subsets of sample spaces, noting that in the sample space: the event “not A” is the complement of A; that the event “A and B” is the intersection of sets A and B; and that the event “A or B” is the union of A and B. The Addition Rule is developed and used to compute a probability. During this unit students examine situations involving pairs of independent and non-independent events, learning to distinguish between such pairs in everyday situations. 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 observe and formalize relationships such as P(A|B) = P(A and B)/P(B), and, when A and B are independent, P(A) = P(A|B). In a previous statistics unit, students encountered two-way frequency tables. In this unit, students revisit these tables as a way to approximate conditional probabilities by treating the tables as a sample space and extend their prior work to consider independence.