Homework Assignment: Probability and Statistical Analysis

This document provides solutions to a statistics and probability homework assignment. The assignment covers various concepts, including calculating the probability of independent events, such as strikes by firefighters and police officers, and the probability of events occurring or not occurring. It explores the concept of mutually exclusive events using examples like coin tosses. The assignment includes problems related to analyzing data from the Central Intelligence Agency (CIA) on the Russian wheat harvest and calculating probabilities based on the data. Additionally, it involves analyzing crime statistics from a survey of households and calculating conditional probabilities and joint probabilities of reported and unreported crimes. The solutions demonstrate the application of probability principles to real-world scenarios.

Problems
7.2 David Morgan, the city manager of Yukon, Oklahoma, must negotiate new
contracts with both the firefighters and the police officers. He plans to offer both
groups a 7% wage increase and hold firm. Mr. Morgan feels that there is one
chance in three that the firefighters will strike and one chance in seven that the
police will strike. Assume that the events are independent.
a. What is the probability that both will strike?
P(Both will strike) = P(Firefighters will strike)*P(Police officers will strike)
= (1/3)*(1/7) = 1/21
b. What is the probability that neither the police nor the firefighters will strike?
P(Neither will strike) = P(Firefighters will not strike)*P(Police officers will not strike)
= (2/3)*(6/7) = 4/7
c. What is the probability that the police will strike and the firefighters will not?
P(police will strike and the firefighters will not)
= P(police will strike)*P(Firefighters will not strike)
=(1/7)*(2/3) = 2/21
d. What is the probability that the firefighters will strike and the police will not?
P(firefighters will strike and the police will not)
= P(firefighters will strike)*P(police will not strike)
=(1/3)*(6/7) = 2/7
7.4 In your own words, explain what mutually exclusive events are.

Mutually exclusive events are two events where there is no probability of two ever occurring at the
same time. A good example is in a coin toss. There isn’t a possibility of getting both heads and tails at
the same time.
7.6 The National Crop Reporting Service receives the accompanying information from the Central
Intelligence Agency (CIA) on the Russian wheat harvest.
Russian demand for wheat is approximately 210 million metric tons.
(a) What is the probability that the harvest will be adequate?
= (0.15+0.07) / 2
= 0.11
= 11%
(b) If U.S. exports make up the difference, what is the probability that the United
States will have to sell more than 20 million metric tons to the Russians?
= 0.05+0.15+0.23
= 0.43
= 43%
Millions of Metric Tons Probability
Less than 170 .05
170–180 .15
180–190 .23
190–200 .31
200–210 .15
210–220 .07
More than 220 .04
7.8 From a survey of the households in Grayson, Missouri, the crime figures shown
in the accompanying table were estimated.
Crime Number Reported Number Not Reported Total
Murder 12 12 24
Robbery 145 105 250
Assault 85 177 262

Rape 12 60 72
Auto theft 314 62 376
Total 568 416 984
(a) What is the probability that a crime is reported?
= 568/984
=0.58
=58%
(b) What is the probability that a crime is reported, given that an assault occurs?
= 85/262
= 0.32
= 32%
(c) What is the probability that a nonreported crime is either a robbery or an
assault?
= (105/250) * (177/262)
= 0.42 * 0.68
= 0.29
= 29%
(c) What is the probability that a crime is a rape and that it is reported?
= (72/984) * (568/984)
= 0.07 * 0.58
= 0.04
= 4%
(e) What is the probability that a crime is reported or that the crime is a robbery?
= (568/984) + (250 / 984)
= 0.58 + 0.25
= 0.83
= 83%