1. Section 5.1
Review and Preview

2. Review and Preview
This chapter combines the methods of descriptive statistics presented in Chapter 2 and 3 and those of probability presented in Chapter 4 to describe and analyze probability distributions. Probability Distributions describe what will probably happen instead of what actually did happen, and they are often given in the format of a graph, table, or formula.
Emphasize the combination of the methods in Chapter 3 (descriptive statistics) with the methods in Chapter 4 (probability).
Chapter 3 one would conduct an actual experiment and find and observe the mean, standard deviation, variance, etc. and construct a frequency table or histogram
Chapter 4 finds the probability of each possible outcome
Chapter 5 presents the possible outcomes along with the relative frequencies we expect
Page 200 of Elementary Statistics, 10th Edition

3. Preview
In order to fully understand probability distributions, we must first understand the concept of a random variable, and be able to distinguish between discrete and continuous random variables. In this chapter we focus on discrete probability distributions. In particular, we discuss binomial probability distributions.
Emphasize the combination of the methods in Chapter 3 (descriptive statistics) with the methods in Chapter 4 (probability).
Chapter 3 one would conduct an actual experiment and find and observe the mean, standard deviation, variance, etc. and construct a frequency table or histogram
Chapter 4 finds the probability of each possible outcome
Chapter 5 presents the possible outcomes along with the relative frequencies we expect
Page 200 of Elementary Statistics, 10th Edition

4. Combining Descriptive Methods
and Probabilities
In this chapter we will construct probability distributions by presenting possible outcomes along with the relative frequencies we expect.
page 200 of Elementary Statistics, 10th Edition

5. Section 5.2
Random Variables

6. Random Variable Probability Distribution
Random variable: a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure. EX: the number of peas with green pods among 5 offspring peas Probability distribution: a description that gives the probability for each value of the random variable; often expressed in the format of a graph, table, or formula.
page 201 of Elementary Statistics, 10th Edition

7. Discrete and Continuous Random Variables
Discrete random variable: either a finite number of values or countable number of values, where “countable” refers to the fact that there might be infinitely many values, but they result from a counting process. *cannot be a decimal!!!
Continuous random variable: infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions.
*could be a decimal!!!
This chapter deals exclusively with discrete random variables - experiments where the data observed is a ‘countable’ value. Give examples.
Following chapters will deal with continuous random variables.
Page 201 of Elementary Statistics, 10th Edition

8. Example 1: Identify the given random variable as being discrete or continuous. a) The number of people now driving a car in the United States. b) The weight of the gold stored in Fort Knox. c) The height of the last airplane departed from JFK Airport in New York City.
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9. Example 1 continued: Identify the given random variable as being discrete or continuous. d) The number of cars in San Francisco that crashed last year. e) The time required to fly from Los Angeles to Shanghai.
page 201 of Elementary Statistics, 10th Edition

10. Requirements for Probability Distribution
The sum of all probabilities is 1.
ΣP(x) = 1, where x assumes all possible values.
(values such as or are acceptable because they result from rounding errors)
Each individual probability is a value between 0 and
1 inclusive.
0  P(x)  1, for every individual value of x.
Page 203 of Elementary Statistics, 10th Edition

11. Probability Distribution
Formulas for a
Probability Distribution
µ = Σ [x • P(x)] Mean σ = Σ[x2 • P(x)] – µ2 Standard Deviation σ2 = Σ[(x – µ)2 • P(x)] Variance σ 2 = Σ[x2 • P(x)] – µ2 Variance (shortcut)
In Chapter 3, we found the mean, standard deviation,variance, and shape of the distribution for actual observed experiments.
The probability distribution and histogram can provide the same type information.
These formulas will apply to ANY type of probability distribution as long as you have have all the P(x) values for the random variables in the distribution.
In section 4 of this chapter, there will be special EASIER formulas for the special binomial distribution.
The TI-83 and TI-83 Plus calculators can find the mean, standard deviation, and variance in the same way that one finds those values for a frequency table.
With the TI-82, TI-81, and TI-85 calculators, one would have to multiply all decimal values in the P(x) column by the same factor so that there were no decimals and proceed as usual.
Page 204 of Elementary Statistics, 10th Edition

12. Roundoff Rule for µ, , and  2
Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round µ, σ, and σ2 to one decimal place.
If the rounding results in a value that looks as if the mean is ‘all’ or none’ (when, in fact, this is not true), then leave as many decimal places as necessary to correctly reflect the true mean.
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13. Example 2: Determine whether or not a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirement(s) that are not satisfied. Three males with X-linked genetic disorder have one child each. The random variable x is the number of children among the three who inherit the X-linked genetic disorder. x
P(x)
𝒙∙𝑷(𝒙)
𝒙 𝟐
𝒙 𝟐 ∙𝑷(𝒙)
0.125 1
0.375 2 3
page 201 of Elementary Statistics, 10th Edition

14. Example 3: Determine whether or not a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirement(s) that are not satisfied. Air America has a policy of routinely overbooking flights. The random variable x represents the number of passengers who cannot be boarded because there are more passengers than seats (based on data from an IBM research paper by Lawrence, Hong, and Cherrier.) x
P(x)
0.051 1
0.141 2
0.274 3
0.331 4
0.187
page 201 of Elementary Statistics, 10th Edition

15. Example 4: Determine whether or not a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirement(s) that are not satisfied. x
P(x) 1
0.6 2
0.2 3 4
0.15 5
0.05
page 201 of Elementary Statistics, 10th Edition

16. Identifying Unusual Results
Range Rule of Thumb
According to the range rule of thumb, most values should lie within 2 standard deviations of the mean.
We can therefore identify “unusual” values by determining if they lie outside these limits:
Maximum usual value = μ + 2σ
Minimum usual value = μ – 2σ
Page of Elementary Statistics, 10th Edition

17. Identifying Unusual Results
Probabilities
Rare Event Rule for Inferential Statistics
If, under a given assumption (such as the assumption that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is probably not correct.
The discussion of this topic in the text includes some difficult concepts, but it also includes an extremely important approach used often in statistics.
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18. Identifying Unusual Results
Probabilities
Using Probabilities to Determine When Results Are Unusual
Unusually high: x successes among n trials is an unusually high number of successes if
P(x or more) ≤ 0.05.
The discussion of this topic in the text includes some difficult concepts, but it also includes an extremely important approach used often in statistics.
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Unusually low: x successes among n trials is an unusually low number of successes if
P(x or fewer) ≤ 0.05.

19. Example 5: Refer to the table, which describes results from eight offspring peas. The random variable x represents the number of offspring peas with green pods. a) Find the probability of getting exactly 7 peas with green pods. b) Find the probability of getting 7 or more peas with green pods. c) Which probability is relevant for determining whether 7 is an unusually high number of peas with green pods: the result from part (a) or part (b)? d) Is 7 an unusually high number of peas with green pods? Why or why not? x
P(x) 0+ 1 2
0.004 3
0.023 4
0.087 5
0.208 6
0.311 7
0.267 8
0.100
page 201 of Elementary Statistics, 10th Edition

20. Example 6: Based on past results found in the Information Please Almanac, there is a probability that a baseball World Series contest will last four games, a probability that it will last five games, a probability that it will last six games, and a probability that it will last seven games.
a) Does the given information describe a probability distribution?
c) Is it unusual for a team to “sweep” by winning in four games? Why or why not?
page 201 of Elementary Statistics, 10th Edition

21. b) Assuming that the given information describes a probability distribution, find the mean and standard deviation for the numbers of games in a World Series contest. x
P(x)
𝒙∙𝑷(𝒙)
𝒙 𝟐
𝒙 𝟐 ∙𝑷(𝒙) 4
0.1919 5
0.2121 6
0.2222 7
0.3737 x
P(x)
𝒙∙𝑷(𝒙)
𝒙 𝟐
𝒙 𝟐 ∙𝑷(𝒙) 4
0.1919 5
0.2121 6
0.2222 7
0.3737

22. Example 8
Let the random variable x represent the number of girls in a family of three children. Construct a table describing the probability distribution, then find the mean and standard deviation. (HINT: List the different possible outcomes.)
Is it unusual for a family of three children to consist of three girls?

23. Different possible outcomes of having three children:
0 girls:
1 girl:
2 girls:
3 girls:
BBB
Mean:
Standard Deviation:
GBB BGB BBG
GGB GBG BGG
GGG x
P(x)
𝒙∙𝑷(𝒙)
𝒙 𝟐
𝒙 𝟐 ∙𝑷(𝒙) 1 2 3
Is it unusual for a family of three children to consist of three girls?

24. GAME TIME! You and your neighbor are going to play a game of rolling a die. The rules of the game are as follows: You and your neighbor each roll 1 die, one time. If you roll a number less than 5, then your neighbor gives you $2. If you roll a 5 or 6, then you give your neighbor $4. Does this seem fair?

25. E = Σ[x • P(x)] Expected Value
The expected value of a discrete random variable is denoted by E, and it represents the mean value of the outcomes. It is obtained by finding the value of Σ [x • P(x)].
E = Σ[x • P(x)]
Also called expectation or mathematical expectation
Plays a very important role in decision theory
page 208 of Elementary Statistics, 10th Edition

26. Example 9: In the Illinois Pick 3 lottery game, you pay 50¢ to select a sequence of three digits, such as 233. If you select the same sequence of three digits that are drawn, you win and collect $250. a) How many different selections are possible? b) What is the probability of winning? c) If you win, what is your net profit?
page 201 of Elementary Statistics, 10th Edition

27. Example 9 continued: In the Illinois Pick 3 lottery game, you pay 50¢ to select a sequence of three digits, such as 233. If you select the same sequence of three digits that are drawn, you win and collect $250. d) Find the expected value. e) If you bet 50 ¢ in Illinois’ Pick 4 game, the expected value is –25¢. Which bet is better: A 50¢ bet in the Illinois Pick 3 game or a 50¢ bet in the Illinois Pick 4 game? Explain.
page 201 of Elementary Statistics, 10th Edition

28. Example 10: When playing roulette at the Bellagio casino in Las Vegas, a gambler is trying to decide whether to bet $5 on the number 13 or to bet $5 that the outcome is any one of these five possibilities: 0 or 00 or 1 or 2 or 3. The expected value for a $5 bet for a single number is –26¢. For the $5 bet that the outcome is 0 or 00 or 1 or 2 or 3, there is a probability of 5/38 of making a net profit of $30 and a 33/38 probability of losing $5. a) Find the expected value for the $5 bet that the outcome is 0 or 00 or 1 or 2 or 3.
page 201 of Elementary Statistics, 10th Edition

29. Example 10 continued: When playing roulette at the Bellagio casino in Las Vegas, a gambler is trying to decide whether to bet $5 on the number 13 or to bet $5 that the outcome is any one of these five possibilities: 0 or 00 or 1 or 2 or 3. The expected value for a $5 bet for a single number is –26¢. For the $5 bet that the outcome is 0 or 00 or 1 or 2 or 3, there is a probability of 5/38 of making a net profit of $30 and a 33/38 probability of losing $5. b) Which bet is better: A $5 bet on the number 13 or a $5 bet that the outcome is 0 or 00 or 1 or 2 or 3? Why?
page 201 of Elementary Statistics, 10th Edition