Probability and Statistical Inference. Robert Bartoszynski

      Associativity:

      Distributivity:

      De Morgan's Laws:

(1.4)

It is often helpful to use Venn diagrams for studying relations between composite events in the same sample space. The sample space is there represented by a rectangle, while its subsets represent events (see Figure 1.5).

Figure 1.5 Complement, union, and intersection.

The complement of event is represented in Figure 1.5a, the union and intersection of the events and are represented in Figure 1.5b and c, respectively.

Venn diagrams can also be used to check the validity of formulas. For example, consider the first De Morgan's law (1.4) for the case of two events:

(1.5)

Venn diagrams made separately for the left‐hand side and the right‐hand side of (1.5) (see Figure 1.6) indicate that both regions are the same. Although a picture does not constitute a proof, it may provide convincing evidence that the statement is true, and sometimes may even suggest a method of proving the statement.

Figure 1.6 The first De Morgan's law.

      Problems

      For the problems below, remember that a statement (expressed as a sentence or formula) is true if it is true under all circumstances, and it is false if there is at least one case where it does not hold.

      1 1.3.1 Answer true or false. Justify your answer. (i) If and are distinct events (i.e., ) such that and are disjoint, then and are also disjoint. (ii) If and are disjoint, then and are also disjoint. (iii) If and are disjoint, and also and are disjoint, then and are disjoint. (iv) If and are both contained in , then . (v) If is contained in , is contained in and is disjoint from , then is disjoint from . (vi) If , then .

      2 1.3.2 In the statements below , and are events. Find those statements or formulas that are true. (i) If , then . (ii) . (iii) . (iv) If , then . (v) . (vi) . (vii) . (viii) If , and , then . (ix) If , and are not empty, then is not empty. (x) Show that .

      3 1.3.3 Find if: (i) . (ii) . (iii) . (iv) .

      4 1.3.4 In a group of 1,000 students of a certain college, 60 take French, 417 take calculus, and 509 take statistics. Moreover, 20 take French and calculus, 17 take French and statistics, and 147 take statistics and calculus. However, 196 students do not take any of these three subjects. Determine the number of students who take French, calculus, and statistics.

      5 1.3.5 Let , and be three events. Match, where possible, events through with events through . Matching means that the events are exactly the same; that is, if one occurs, so must the other and conversely (see the Definition 1.3.2). (Hint: Draw a Venn diagram for each event do the same for events , and then compare the diagrams.)Among events , , : two or more occur. exactly one occurs. only occurs. all occur. none occurs. at most one occurs. at least one occurs. exactly two occur. no more than two occur. occurs.

      6 1.3.6 A standard deck of cards is dealt among players and . Let be the event “ has at least aces,” and let , and be defined similarly. For each of the events below, determine the number of aces that has. (i) . (ii) . (iii) . (iv) . (v) . (vi) .

      7 1.3.7 Five burglars, and , divide the loot, consisting of five identical gold bars and four identical diamonds. Let be the event that got at least gold bars and at most diamonds. Let denote analogous events for burglars (e.g., is the event that got 2, 3, 4, or 5 gold bars and 0 or 1 diamond). Determine the number of gold bars and the number of diamonds received by if the following events occur (if determination of and/or is impossible, give the range of values): (i) . (ii) . (iii) . (iv) .

      8 1.3.8 Let be defined inductively by . Find and for .

1.4 Infinite Operations on Events

      As already mentioned, the operations of union and intersection can be extended to infinitely many events. Let be an infinite sequence of events. Then,

      are events “at least one occurs” and “all 's occur,” respectively.

If at least one event occurs, then there is one that occurs first. This remark leads to the following useful decomposition of a union of events into a union of disjoint events:

(1.6)

      where is the event “ is the first event in the sequence that occurs.”

      For an infinite sequence one can define two events:

(1.7)

      and

      (1.8)

      these