EXERCISES Section 2.5 (70-89)

70

Reconsider the credit card scenario of Exercise 47 (Section 2.4), and show that $A$ and $B$ are dependent first by using the definition of independence and then by verifying that the multiplication property does not hold.

71

An oil exploration company currently has two active projects, one in Asia and the other in Europe. Let $A$ be the event that the Asian project is successful and $B$ be the event that the European project is successful. Suppose that $A$ and $B$ are independent events with $P\left(A\right)=.4$ and $P\left(B\right)=.7$.

1. If the Asian project is not successful, what is the probability that the European project is also not successful? Explain your reasoning.
2. What is the probability that at least one of the two projects will be successful?
3. Given that at least one of the two projects is successful, what is the probability that only the Asian project is successful?

72

In Exercise 13, is any $A_i$ independent of any other $A_j$ ? Answer using the multiplication property for independent events.

73

If $A$ and $B$ are independent events, show that $A^{\prime }$ and $B$ are also independent.

• Hint: First establish a relationship between $P\left(A^{\prime }\cap B\right),P\left(B\right)$ , and $P\left(A\cap B\right)$.

74

The proportions of blood phenotypes in the U.S. population are as follows:

A	B	AB	O
.40	.11	.04	.45

Assuming that the phenotypes of two randomly selected individuals are independent of one another, what is the probability that both phenotypes are $\mathrm{O}$ ? What is the probability that the phenotypes of two randomly selected individuals match?

75

One of the assumptions underlying the theory of control charting (see Chapter 16) is that successive plotted points are independent of one another. Each plotted point can signal either that a manufacturing process is operating correctly or that there is some sort of malfunction.

Even when a process is running correctly, there is a small probability that a particular point will signal a problem with the process. Suppose that this probability is.05. What is the probability that at least one of 10 successive points indicates a problem when in fact the process is operating correctly? Answer this question for 25 successive points.

76

In October, 1994, a flaw in a certain Pentium chip installed in computers was discovered that could result in a wrong answer when performing a division. The manufacturer initially claimed that the chance of any particular division being incorrect was only 1 in 9 billion, so that it would take thousands of years before a typical user encountered a mistake. However, statisticians are not typical users; some modern statistical techniques are so computationally intensive that a billion divisions over a short time period is not outside the realm of possibility. Assuming that the 1 in 9 billion figure is correct and that results of different divisions are independent of one another, what is the probability that at least one error occurs in one billion divisions with this chip?

77

An aircraft seam requires 25 rivets. The seam will have to be reworked if any of these rivets is defective. Suppose rivets are defective independently of one another, each with the same probability.

1. If $15\%$ of all seams need reworking, what is the probability that a rivet is defective?
2. How small should the probability of a defective rivet be to ensure that only $10\%$ of all seams need reworking?

78

A boiler has five identical relief valves. The probability that any particular valve will open on demand is.96. Assuming independent operation of the valves, calculate $P$ (at least one valve opens) and $P$ (at least one valve fails to open).

79

Two pumps connected in parallel fail independently of one another on any given day. The probability that only the older pump will fail is.10 , and the probability that only the newer pump will fail is.05. What is the probability that the pumping system will fail on any given day (which happens if both pumps fail)?

80

Consider the system of components connected as in the accompanying picture. Components 1 and 2 are connected in parallel, so that subsystem works iff either 1 or 2 works; since 3 and 4 are connected in series, that subsystem works iff both 3 and 4 work. If components work independently of one another and $P$ (component $i$ works) $=.9$ for $i=1,2$ and $=.8$ for $i=3,4$ , calculate $P$ (system works).

81

Refer back to the series-parallel system configuration introduced in Example 2.36, and suppose that there are only two cells rather than three in each parallel subsystem [in Figure 2.14(a), eliminate cells 3 and 6, and renumber cells 4 and 5 as 3 and 4]. Using $P\left(A_i\right)=.9$ , the probability that system lifetime exceeds $t_0$ is easily seen to be.9639. To what value would. 9 have to be changed in order to increase the system lifetime reliability from.9639 to.99 ?

• Hint: Let $P\left(A_i\right)=p$ , express system reliability in terms of $p$ , and then let $x=p^2$.

82

Consider independently rolling two fair dice, one red and the other green. Let

• $A$ be the event that the red die shows 3 dots,
• $B$ be the event that the green die shows 4 dots,
• $C$ be the event that the total number of dots showing on the two dice is 7.

Are these events pairwise independent (i.e., are $A$ and $B$ independent events, are $A$ and $C$ independent, and are $B$ and $C$ independent)? Are the three events mutually independent?

83

Components arriving at a distributor are checked for defects by two different inspectors (each component is checked by both inspectors). The first inspector detects $90\%$ of all defectives that are present, and the second inspector does likewise. At least one inspector does not detect a defect on $20\%$ of all defective components. What is the probability that the following occur?

A defective component will be detected only by the first inspector? By exactly one of the two inspectors?

All three defective components in a batch escape detection by both inspectors (assuming inspections of different components are independent of one another)?

84

Consider purchasing a system of audio components consisting of a receiver, a pair of speakers, and a CD player. Let $A_1$ be the event that the receiver functions properly throughout the warranty period, $A_2$ be the event that the speakers function properly throughout the warranty period, and $A_3$ be the event that the CD player functions properly throughout the warranty period. Suppose that these events are (mutually) independent with $P\left(A_1\right)=$ $.95,P\left(A_2\right)=.98$ , and $P\left(A_3\right)=.80$.

What is the probability that all three components function properly throughout the warranty period?

What is the probability that at least one component needs service during the warranty period?

What is the probability that all three components need service during the warranty period?

What is the probability that only the receiver needs service during the warranty period?

What is the probability that exactly one of the three components needs service during the warranty period?

What is the probability that all three components function properly throughout the warranty period but that at least one fails within a month after the warranty expires?

85

A quality control inspector is examining newly produced items for faults. The inspector searches an item for faults in a series of independent fixations, each of a fixed duration. Given that a flaw is actually present, let $p$ denote the probability that the flaw is detected during any one fixation (this model is discussed in “Human Performance in Sampling Inspection,” Human Factors, 1979: 99-105).

Assuming that an item has a flaw, what is the probability that it is detected by the end of the second fixation (once a flaw has been detected, the sequence of fixations terminates)?

Give an expression for the probability that a flaw will be detected by the end of the $n$ th fixation.

If when a flaw has not been detected in three fixations, the item is passed, what is the probability that a flawed item will pass inspection?

Suppose $10\%$ of all items contain a flaw $[P$ (randomly chosen item is flawed) $=.1]$. With the assumption of part (c), what is the probability that a randomly chosen item will pass inspection (it will automatically pass if it is not flawed, but could also pass if it is flawed)?

Given that an item has passed inspection (no flaws in three fixations), what is the probability that it is actually flawed? Calculate for $p=.5$.

86

A lumber company has just taken delivery on a shipment of $10,0002\times 4$ boards. Suppose that $20\%$ of these boards (2000) are actually too green to be used in first-quality construction. Two boards are selected at random, one after the other. Let $A=\{$ the first board is green $\}$ and $B=\{$ the second board is green $\}$. Compute $P\left(A\right),P\left(B\right)$ , and $P\left(A\cap B\right)$ (a tree diagram might help). Are $A$ and $B$ independent?

With $A$ and $B$ independent and $P\left(A\right)=P\left(B\right)=.2$ , what is $P\left(A\cap B\right)$ ? How much difference is there between this answer and $P\left(A\cap B\right)$ in part (a)? For purposes of calculating $P\left(A\cap B\right)$ , can we assume that $A$ and $B$ of part (a) are independent to obtain essentially the correct probability?

Suppose the shipment consists of ten boards, of which two are green. Does the assumption of independence now yield approximately the correct answer for $P\left(A\cap B\right)$ ? What is the critical difference between the situation here and that of part (a)? When do you think an independence assumption would be valid in obtaining an approximately correct answer to $P\left(A\cap B\right)$ ?

87

Consider randomly selecting a single individual and having that person test drive 3 different vehicles. Define events ${\mathrm{A}}_1,{\mathrm{A}}_2$ , and ${\mathrm{A}}_3$ by

$A_1=$ likes vehicle $\#1A_2=$ likes vehicle $\#2$

$A_3=$ likes vehicle #3

Suppose that $P\left(A_1\right)=.55,P\left(A_2\right)=.65,P\left(A_3\right)=.70$ ,

$P\left(A_1\cup A_2\right)=.80,P\left(A_2\cap A_3\right)=.40$ , and

$P\left(A_1\cup A_2\cup A_3\right)=.88$ 。

1. What is the probability that the individual likes both vehicle #1 and vehicle #2?

2. Determine and interpret $P\left(A_2\mid A_3\right)$.

3. Are ${\mathrm{A}}_2$ and ${\mathrm{A}}_3$ independent events? Answer in two different ways.

4. If you learn that the individual did not like vehicle #1 , what now is the probability that he/she liked at least one of the other two vehicles?

88

The probability that an individual randomly selected from a particular population has a certain disease is .05. A diagnostic test correctly detects the presence of the disease $98\%$ of the time and correctly detects the absence of the disease $99\%$ of the time. If the test is applied twice, the two test results are independent, and both are positive, what is the (posterior) probability that the selected individual has the disease?

• Hint: Tree diagram with first-generation branches corresponding to Disease and No Disease, and second- and third-generation branches corresponding to results of the two tests.

89

Suppose identical tags are placed on both the left ear and the right ear of a fox. The fox is then let loose for a period of time. Consider the two events $C_1=\{$ left ear tag is lost $\}$ and $C_2=\{$ right ear tag is lost $\}.$ Let $\pi =P\left(C_1\right)=P\left(C_2\right)$ , and assume $C_1$ and $C_2$ are independent events. Derive an expression (involving $\pi$ ) for the probability that exactly one tag is lost, given that at most one is lost (“Ear Tag Loss in Red Foxes,” J. Wildlife Mgmt., 1976: 164-167).

• Hint: Draw a tree diagram in which the two initial branches refer to whether the left ear tag was lost.