math 1105: homework 2 solutions

Math 1105: Homework 2 Solutions

Part A

7.3

11. An event is a subset of a sample space.

7.4

1. Two events are mutually exclusive if it is impossible for both of them to happen at thesame time. Equivalently, events E and F are mutually exclusive if the intersection E ∩ F is

the empty set.

2. Union rule for mutually exclusive events: If E and F are mutually exclusive events, thenP (E ∪ F ) = P (E) + P (F ). This is because the event E ∪ F occurs exactly when either Eor F occurs. Ordinarily, summing P (E)+P (F ) would double-count all the outcomes in theintersection E ∩ F ; but since E and F are mutually exclusive, there are no such outcomesto double-count.

Part B

7.3

7.4

45. This is a trick question. Let B be the event that Linda is a bank teller, and let F be theevent that Linda is a feminist. You are given a lot of information indicating that P (F ) isvery high (close to 1). However, the question is: Which is more likely, that Linda is a bankteller (the event B) or that Linda is a bank teller and a feminist (the event B ∩F )? Clearly,if Linda is both a bank teller and a feminist, so that B ∩F is true, then in particular she is abank teller (that is, B is true). In the language of set theory, B ∩ F is a subset of B. ThusP (B ∩ F ) must be less than or equal to P (B). This is true no matter how likely F is.

7.4