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Statistics The Art and Science of Learning from Data

Alan Agresti, Christine A. Franklin, Bernhard Klingenberg

Chapter 5

Probability in Our Daily Lives - all with Video Answers

Educators

Section 1

How Probability Quantifies Randomness

00:43

Problem 1

Explain what is meant by the long-run relative frequency definition of probability.

Christopher Stanley
Christopher Stanley
Numerade Educator
00:19

Problem 2

The objective of the game Minesweeper is to clear a field without detonating any mines or bombs. Your friend claims that his rate of completing the game successfully is $90 \%$.
a. You decide to challenge your friend. He makes 10 attempts to complete the game, but is successful in only 7 of them. Does this mean that your friend's claim is wrong?
b. If your friend's claim was actually true, what would you have to do to ensure that the cumulative proportion of his successful attempts to complete the game falls very close to $0.9 ?$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:50

Problem 3

You visit your counselor's office at 10 randomly chosen times, and he is not available at any of those times. Does this mean that the probability of your counselor being available at his office for students equals 0 ? Explain.

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
00:42

Problem 4

Airplane safety has been improving over the years. From 2000 to $2010,$ the average number of global airline deaths per year was over 1000 , even when excluding the nearly 3000 deaths in the United States on September 11,2001 . The number of global airline deaths declined in 2011 , again in $2012,$ and then hit a low of only 265 in $2013 .$ In $2013,$ there were a total of 825 million passengers globally. Sources: en.wikipedia.org and www.transtats.bts.gov1.
a. Can you consider the 2013 data as a long run or short run of trials? Explain.
b. Estimate the probability of dying on a flight in $2013 .$ (Note, the probability of dying from a 1000 -mile automobile trip is about 1 in 42,000 by contrast.)
c. Raul is considering flying on an airplane. He noticed that over the past two months, there have been no fatal airplane crashes around the world. This raises his concern about flying because the airlines are "due for an accident." Comment on his reasoning.

Arwa Ali
Arwa Ali
Numerade Educator
07:21

Problem 5

The powerrank.com website (http:// thepowerrank.com/2014/06/06/world-cup-2014-winprobabilities-from-the-power-rank/) listed the probability of each team to win the 2014 World Cup in soccer as follows:
1. Brazil, $35.9 \%$.
2. Argentina, $10.0 \%$.
3. Spain, $8.9 \%$.
4. Germany, $7.4 \%$.
5. Netherlands, $5.7 \%$.
6. Portugal, $3.9 \%$.
7. France, $3.4 \%$.
8. England, $2.8 \%$.
9. Uruguay, $2.5 \%$.
10. Mexico, $2.5 \%$.
11. Italy, $2.3 \%$.
12. Ivory Coast, $2.0 \%$,
13. Colombia, $1.5 \%$.
14. Russia, $1.5 \%$.
15. United States, $1.1 \%$.
16. Chile, $1.0 \%$.
17. Croatia, $0.9 \%$
18. Ecuador, $0.8 \%$.
19. Nigeria, $0.8 \%$.
20. Switzerland, $0.7 \%$.
21. Greece, $0.6 \%$
22. $\operatorname{Iran}, 0.6 \%$.
23. Japan, $0.6 \%$.
24. Ghana, $0.6 \%$.
25. Belgium, $0.4 \%$.
26. Honduras, $0.3 \%$.
27. South Korea, $0.3 \%$.
28. Bosnia-Herzegovina, $0.3 \%$.
29. Costa Rica, $0.3 \%$.
30. Cameroon, $0.2 \%$.
31. Australia, $0.2 \%$.
32. Algeria, $0.1 \%$.
a. Interpret Brazil's probability of $35.9 \%,$ which was based on computer simulations of the tournament. Is it a relative frequency or a subjective interpretation of probability?
b. Germany would emerge as the actual winner of the 2014 World Cup. Does this indicate that the $7.4 \%$ chance of Germany winning, which was calculated before the tournament, should have been $100 \%$ instead?

Evelyn Cunningham
Evelyn Cunningham
Numerade Educator
05:30

Problem 6

Which of the following statements is not correct, and why?
a. If the number of male and female employees at a call center is equal, then the probability that a call is answered by a female employee is 0.50 .
b. If you randomly generate 10 digits, each integer between 0 and 9 must occur exactly once.
c. You have 1,000 songs on your MP3 disc. 150 of them are of your favorite artist. If you decide to randomly play a very large number of songs, then each song of your favorite artist would have been played almost $15 \%$ of the time.

Ahmad Reda
Ahmad Reda
Numerade Educator
00:39

Problem 7

Your friend is interested in estimating the proportion of people who would vote for his project in a local contest. He selects a large sample among his many friends and claims that, with such a large sample, he does not need to worry about the method of selecting the sample. What is wrong in this reasoning? Explain.

Nick Johnson
Nick Johnson
Numerade Educator
02:03

Problem 8

Before the first human heart transplant, Dr. Christiaan Barnard of South Africa was asked to assess the probability that the operation would be successful. Did he need to rely on the relative frequency definition or the subjective definition of probability? Explain.

Qudsiya Anis
Qudsiya Anis
Numerade Educator
01:37

Problem 9

You are asked to use your best judgment to estimate the probability that there will be a nuclear war within the next 10 years. Is this an example of relative frequency or subjective definition of probability? Explain.

Christopher Stanley
Christopher Stanley
Numerade Educator
02:39

Problem 10

Use the web app Random Numbers (go to the tab that says Coin Flips) on the book's website or other software (such as random.org/coin) to illustrate the long-run definition of probability by simulating shortterm and long-term results of flipping a balanced coin.
a. Keep the probability of a head at the default value of $50 \%$ and set the number of flips to generate in a simulation to $10 .$ Click on Simulate and record the proportion of heads for this simulation. Do this a total of 10 times by repeatedly clicking Simulate.
b. Now set the number of flips to $100 .$ Click Simulate 10 times, and record the 10 proportions of heads for each simulation. Do they vary much?
c. Now set the number of flips to $1000 .$ Click Simulate 10 times, and record the 10 proportions of heads for each simulation. Do they vary more, or less, than the proportions in part b based on 100 flips?
d. Summarize the effect of the number of flips on the variability of the proportion. How does this reflect what's implied by the law of large numbers?

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
01:55

Problem 11

A teacher announces a pop quiz for which the student is completely unprepared. The quiz consists of 100 true-false questions. The student has no choice but to guess the answer randomly for all 100 questions.
a. Simulate taking this quiz by random guessing. Number a sheet of paper 1 to 100 to represent the 100 questions. Write a T (true) or F (false) for each question, by predicting what you think would happen if you repeatedly flipped a coin and let a tail represent a T guess and a head represent an F guess. (Don't actually flip a coin; merely write down what you think a random series of guesses would look like.)
b. How many questions would you expect to answer correctly simply by guessing?
c. The table shows the 100 correct answers. The answers should be read across rows. How many questions did you answer correctly?
d. The preceding answers were actually randomly generated by an app. What percentage were true, and what percentage would you expect? Why are they not necessarily identical?
e. Are there groups of answers within the sequence of 100 answers that appear nonrandom? For instance, what is the longest run of Ts or Fs? By comparison, which is the longest run of Ts or Fs within your sequence of 100 answers? (There is a tendency in guessing what randomness looks like to identify too few long runs in which the same outcome occurs several times in a row.)

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 12

An interview in an investment magazine (In the Vanguard, Autumn 2003 ) asked mathematician John Allen Paulos, "What common errors do investors make?" He answered, "People tend not to believe that markets move in random ways. Randomness is difficult to recognize. If you have people write down 100 Hs and Ts to simulate 100 flips of a coin, you will always be able to tell a sequence generated by a human from one generated by real coin flips. When humans make up the sequence, they don't put in enough consecutive Hs and Ts, and they don't make the lengths of those runs long enough or frequent enough. And that is one of the reasons people look at patterns in the stock market and ascribe significance to them." (@) The Vanguard Group, Inc., used with permission.)
a. Suppose that on each of the next 100 business days the stock market has a $1 / 2$ chance of going up and a $1 / 2$ chance of going down, and its behavior one day is independent of its behavior on another day. Use software, such as the web app mentioned in exercise 5.10 or random.org/coin to simulate whether the market goes up or goes down for each of the next 100 days. What is the longest sequence of consecutive moves up or consecutive moves down that you observe?
b. Run the simulation nine more times, with 100 observations for each run, and each time record the longest sequence of consecutive moves up or consecutive moves down that you observe. For the 10 runs, summarize the proportion of times that the longest sequence was $1,2,3,4,5,6,7,8,$ or more. (Your class may combine results to estimate this more precisely.)
c. Based on your findings, explain why if you are a serious investor you should not get too excited if sometime in the next few months you see the stock market go up for five days in a row or go down for five days in a row.

Shu Naito
Shu Naito
Numerade Educator