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Chapter 3: Q108SE (page 204)

Speeding linked to fatal car crashes. According to the National Highway Traffic and Safety Administration’s National Center for Statistics and Analysis (NCSA), “Speeding is one of the most prevalent factors contributing to fatal traffic crashes” (NHTSA Technical Report, August 2005). The probability that speeding is a cause of a fatal crash is .3. Furthermore, the probability that speeding and missing a curve are causes of a fatal crash is .12. Given speeding is a cause of a fatal crash, what is the probability that the crash occurred on a curve?

Short Answer

Expert verified

The probability is 0.4.

Step by step solution

01

Important formula

The formula is P(A∩B)=P(A).P(B|A).

02

The probability that the crash occurred on a curve.

Here, P(G)=0.3=Speeding is a cause of a fatal crash.

P(F|G)=0.12=Speeding and missing a curve are causes of a fatal crash.

P(G∩F)=P(G)×P(F|G)0.12=(0.38)P(F|G)P(F|G)=0.4

Therefore the probability is 0.4.

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Most popular questions from this chapter

Suppose the events B1and B2are mutually exclusive and complementary events, such thatP(B1)=.75andP(B2)=.25 Consider another event A such that role="math" localid="1658212959871" P(AB1)=.3, role="math" localid="1658213029408" P(AB2)=.5.

  1. FindP(B1∩A).
  2. FindP(B2∩A)
  3. Find P(A) using part a and b.
  4. Findrole="math" localid="1658213127512" P(B1A).
  5. Findrole="math" localid="1658213164846" P(B2A).

Reliability of gas station air gauges. Tire and automobile manufacturers and consumer safety experts all recommend that drivers maintain proper tire pressure in their cars. Consequently, many gas stations now provide air pumps and air gauges for their customers. In a Research Note(Nov. 2001), the National Highway Traffic Safety Administration studied the reliability of gas station air gauges. The next table gives the percentage of gas stations that provide air gauges that over-report the pressure level in the tire.

a. If the gas station air pressure gauge reads 35 psi, what is the probability that the pressure is over-reported by 6 psi or more?

b. If the gas station air pressure gauge reads 55 psi, what is the probability that the pressure is over-reported by 8 psi or more?

c. If the gas station air pressure gauge reads 25 psi, what is the probability that the pressure is not over-reported by 4 psi or more?

d. Are the events A= {over report by 4 psi or more} and B= {over report by 6 psi or more} mutually exclusive?

e.Based on your answer to part d, why do the probabilities in the table not sum to 1?

In a random sample of 106 social (or service) robots designed to entertain, educate, and care for human users, 63 were built with legs only, 20 with wheels only, 8 with both legs and wheels, and 15 with neither legs nor wheels. One of the 106 social robots is randomly selected and the design (e.g., wheels only) is noted.

  1. List the sample points for this study.
  2. Assign reasonable probabilities to the sample points.
  3. What is the probability that the selected robot is designed with wheels?
  4. What is the probability that the selected robot is designed with legs?

The outcomes of two variables are (Low, Medium, High) and (On, Off), respectively. An experiment is conducted in which the outcomes of each of the two variables are observed. The accompanying two-way table gives the probabilities associated with each of the six possible outcome pairs.

Low

Medium

High

On

.50

.10

.05

Off

.25

.07

.03

Consider the following events:

A: {On}

B: {Medium or on}

C: {Off and Low}

D: {High}

a. Find P (A).

b. Find P (B).

c. Find P (C).

d. Find P (D).

e. FindP(AC).

f. FindP(A∪B).

g. FindP(A∩B).

h. Consider each pair of events (A and B, A and C, A and D, B and C, B and D, C and D). List the pairs of events that are mutually exclusive. Justify your choices.

Ambulance response time.Geographical Analysis(Jan. 2010) presented a study of emergency medical service (EMS) ability to meet the demand for an ambulance. In one example, the researchers presented the following scenario. An ambulance station has one vehicle and two demand locations, A and B. The probability that the ambulance can travel to a location in under 8 minutes is .58 for location A and .42 for location B. The probability that the ambulance is busy at any point in time is .3.

a.Find the probability that EMS can meet the demand for an ambulance at location A.

b.Find the probability that EMS can meet the demand for an ambulance at location B.
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