Prove Theorem 2.2.2 on page 71. The only if direction is direct from the definition of independence on page 68. For the if direction, use

Prove Theorem 2.2.2 on page 71. The €œonly if€ direction is direct from the definition of independence on page 68. For the €œif€ direction, use induction on the value of j in the definition of independence. Let m = j ˆ’ 1 and let = 1 with j1 = ij.
In Theorem 2.2.2
Let A1, . . . , Ak be events such that Pr(A1 ˆ© . . . ˆ© Ak) > 0. Then A1, . . . , Ak are independent if and only if, for every two disjoint subsets {i1, . . . , im} and {j1, . . . , je} of {1, . . . , k}, we have

Theorem 2.2.2 says that k events are independent if and only if learning that some of the events occur does not change the probability that any combination of the other events occurs.

Pr(A;, n..NAIA,n..nA) = Pr(A, n.n A).