1.3 Problems on probability systems  (Page 2/2)

A committee of five is chosen from a group of 20 people. What is the probability that a specified member of the group will be on the committee?

Ten employees of a company drive their cars to the city each day and park randomly in ten spots. What is the (classical) probability that on a given day Jim willbe in place three? There are $n!$ equally likely ways to arrange n items (order important).

An extension of the classical model involves the use of areas. A certain region L (say of land) is taken as a reference. For any subregion A , define $P\left(A\right)=area\left(A\right)/area\left(L\right)$ . Show that $P(\cdot )$ is a probability measure on the subregions of L .

John thinks the probability the Houston Texans will win next Sunday is 0.3 and the probability the Dallas Cowboys will win is 0.7 (they are not playing each other).He thinks the probability both will win is somewhere between—say, 0.5. Is that a reasonable assumption? Justify your answer.

Suppose $P\left(A\right)=0.5$ and $P\left(B\right)=0.3$ . What is the largest possible value of $P\left(AB\right)$ ? Using the maximum value of $P\left(AB\right)$ , determine $P\left(A{B}^c\right)$ , $P\left(A^{c}B\right)$ , $P\left(A^c{B}^c\right)$ and $P(A\cup B)$ . Are these values determined uniquely?

For each of the following probability “assignments”, fill out the table. Which assignments are not permissible? Explain why, in each case.

The class $\{A,B,C\}$ of events is a partition. Event A is twice as likely as C and event B is as likely as the combination A or C . Determine the probabilities $P\left(A\right),P\left(B\right),P\left(C\right)$ .

Determine the probability $P(A\cup B\cup C)$ in terms of the probabilities of the events $A,B,C$ and their intersections.

If occurrence of event A implies occurrence of B , show that $P\left(A^{c}B\right)=P\left(B\right)-P\left(A\right)$ .

Show that $P\left(AB\right)\ge P\left(A\right)+P\left(B\right)-1$ .

The set combination $A\oplus B=A{B}^c\bigvee A^{c}B$ is known as the disjunctive union or the symetric difference of A and B . This is the event that only one of the events A or B occurs on a trial. Determine $P(A\oplus B)$ in terms of $P\left(A\right)$ , $P\left(B\right)$ , and $P\left(AB\right)$ .

Use fundamental properties of probability to show

1. $P\left(AB\right)\le P\left(A\right)\le P(A\cup B)\le P\left(A\right)+P\left(B\right)$
2. ${\displaystyle P\left(\bigcap _{j=1}^{\infty },E_j\right)\le P\left(E_i\right)\le P\left(\bigcup _{j=1}^{\infty },E_j\right)\le \sum _{j=1}^{\infty }P\left(E_j\right)}$

Suppose $P_1,P_2$ are probability measures and $c_1,c_2$ are positive numbers such that $c_1+c_2=1$ . Show that the assignment $P\left(E\right)=c_1{P}_1\left(E\right)+c_2{P}_2\left(E\right)$ to the class of events is a probability measure. Such a combination of probability measures is known as a mixture . Extend this to

Suppose $\{A_1,A_2,\cdots ,A_n\}$ is a partition and $\{c_1,c_2,\cdots ,c_n\}$ is a class of positive constants. For each event E , let

Show that $Q(\cdot )$ us a probability measure.