1.3 Problems on probability systems

Let Ω consist of the set of positive integers. Consider the subsets

$A=\{\omega :\omega \le 12\}B=\{\omega :\omega <8\}C=\{\omega :\omega \text{is}\text{even}\}$

$D=\{\omega :\omega \text{is}\text{a}\text{multiple}\text{of}\text{3}\}E=\{\omega :\omega \text{is}\text{a}\text{multiple}\text{of}\text{4}\}$

Describe in terms of $A,B,C,D,E$ and their complements the following sets:

1. $\{1,3,5,7\}$
2. $\{3,6,9\}$
3. $\{8,10\}$
4. The even integers greater than 12.
5. The positive integers which are multiples of six.
6. The integers which are even and no greater than 6 or which are odd and greater than 12.

Let Ω be the set of integers 0 through 10. Let $A=\{5,6,7,8\}$ , $B=$ the odd integers in Ω , and $C=$ the integers in Ω which are even or less than three. Describe the following sets by listing their elements.

1. $AB$
2. $AC$
3. $A{B}^c\cup C$
4. $AB{C}^c$
5. $A\cup B^c$
6. $A\cup B{C}^c$
7. $ABC$
8. $A^{c}B{C}^c$

Consider fifteen-word messages in English. Let $A=$ the set of such messages which contain the word “bank” and let $B=$ the set of messages which contain the word “bank” and the word “credit.” Which event has the greater probability? Why?

A group of five persons consists of two men and three women. They are selected one-by-one in a random manner. Let E _i be the event a man is selected on the i th selection. Write an expression for the event that both men have been selected by thethird selection.

Two persons play a game consecutively until one of them is successful or there are ten unsuccessful plays. Let E _i be the event of a success on the i th play of the game. Let $A,B,C$ be the respective events that player one, player two, or neither wins. Write an expression for each of these events in terms of the events E _i , $1\le i\le 10$ .

Suppose the game in [link] could, in principle, be played an unlimited number of times. Write an expression for the event D that the game will be terminated with a success in a finite number of times. Write an expression for the event F that the game will never terminate.

Find the (classical) probability that among three random digits, with each digit (0 through 9) being equally likely and each triple equally likely:

1. All three are alike.
2. No two are alike.
3. The first digit is 0.
4. Exactly two are alike.

The classical probability model is based on the assumption of equally likely outcomes. Some care must be shown in analysis to be certain that this assumption is good.A well known example is the following. Two coins are tossed. One of three outcomes is observed: Let ω ₁ be the outcome both are “heads,” ω ₂ the outcome that both are “tails,” and ω ₃ be the outcome that they are different. Is it reasonable to suppose these three outcomes areequally likely? What probabilities would you assign?