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4.0 Discrete Probability Distributions

4.1 Introduction

4.1.1 Discuss the concept of a random variable

4.1.2 Explain the difference between a discrete random variable and a continuous random variable

When the random variable can take on only a finite number of values or a countable number of values, we say that the variable is a discrete random variable.    A discrete random variable can assume only certain clearly separated values resulting from a count of some item of interest.  Example:  the number of highway deaths in Ontario on the Labour Day weekend may be 5, 6, 7 ... etc.  (not 5.3).  This does not mean that the discrete random variable may not assume fraction values - but it does mean that there is some distance between the values.  The result is still in a count - example:  12 stocks increased by $0.25 or 1/4.

When the random variable can take on any number on the number line - not just counting - we say that the variable is a continuous random variable.  We can have 2.5 or 1.8 mm of rainfall for example or in a high school track met, the winning time for the mile run may be reported as 4 minutes 20 seconds; 4 minutes 20.2 seconds; or 4 minutes 20.3416 seconds, etc.  

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NOTE: If the problem involves counting something, the resulting distribution is usually a discrete probability distribution.

If the distribution is the result of a measurement, then it is usually a continuous probability distribution.

Example: State whether the following random variables are discrete or random.

1. the number of letters delivered on time

2. the length of a package

3. the number who attended

4. the height of each student

4.1.3 Show how probability functions assign probabilities to different values of random variables.

The following is a probability function that assigns probabilities to a random variable.

4.2 The mean of a random variable

4.2.1 Show how the mean of a probability function helps to give the expected value of a probability function.

Special Note:  The mean of a probability distribution is often referred to as the "expected value" of the distribution.

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This formula directs you to multiply each outcome (x) by its probability  P(x); and then add the products. 

Example 1: Calculate the mean of the following probability function representing the probabilities of batters getting a given number of strikes.

x P(x)

0 0.2

1 0.35

2 0.15

3 0.3

4.3 Measuring Chance Variation

4.3.1 Show how the variance of a probability function helps to give the spread of a probability function.

The standard deviation of a discrete probability distribution is found by taking the square root of 2, thus square root of 2

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variance of probability distribution: σ 2=∑ [(x−μ)2∗P (x )]

Example 1: Find the variance of the following probability distribution given in 4.2.1.

x P(x)

0 0.2

1 0.35

2 0.15

3 0.3

Example 2: Find the following using the population distribution below.

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a. What is the probability of exactly 2?

b, What is the probability of exactly none?

c. What is the probability of at least 2?

d. What is the probability of more than 3?

e. What is the probability of no more than 1?

f. Calculate the mean.

g. Calculate the variance.

4.4 Binomial Distribution

x P(x)

0 0.1

1 0.15

2 0.2

3 0.4

4 0.15

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4.4.1 Describe a binomial experiment.

One of the most widely used discrete probability distributions is the binomial probably distribution.

Illustrations of each characteristic:

1. An outcome is classified as either a "success" or "failure."  For example, 40 percent of the students at a particular university are enrolled in the Business program.  For a selected student there are only two possible outcomes - the student is enrolled in the program (a success) or is not enrolled in the Business program (failure).

2. The binomial distribution is the result of counting the number of successes in a fixed sample size.  If we select 5 students,  0, 1, 2, 3, 4 or 5 could be enrolled in the Business program.  This rules out the possibility of 3.75 of the students being enrolled.  There cannot be fractional counts.

3. Probability of success stays the same - in this example the probability of a success remains at 40% for all five students selected.

4. Trials are independent - this means that if the first student selected is enrolled in the Business program, it has no effect on whether the second or fourth one selected will be in the Business Program.

4.4.2 State and graph the formula for a binomial experiment.

There will be TWO ways to get the probability values for a binomial experiment. The first way is to use the formula below and the second way is to use a table of values and look up your answer. This saves a LOT of time, but be careful to use the right method. If you are asked to use the formula to come up with a probability then use the following method.

To construct a binomial distribution, we need to know:

1. the number of trials - designated n2. the probability of successes on each trial - designated

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Where:

nCx denotes a combination (i.e. use the Combination formula) n    is the number of trials x    is the number of observed successes is the probability of success on each trial

Examples:

1. There are five flights daily from Halifax. Suppose the probability that any flight arrives late is 0.20. Use the binomial formula to determine the following probabilities.

a. What is the probability that none of the flights are late today?

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b. What is the probability that exactly one of the five flights will arrive late today?

c. What is the probability that exactly three of the five flights will arrive late today?

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4.4.3 Solve Binomial Distribution Problems using the binomial distribution tables.

Unless it states that you must use the binomial distribution formula, USE THE TABLES! They are sooooo much easier and quicker to use. There are a bunch of tables. To figure out which one you need you have to know your n value: which represents the number of trials. You also need to know your π value (your probability of success on each trial)

Examples:

1. Five percent of the gears produced by a machine are defective.

a. What is the probability that out of 6 gears selected at random, none will be defective?

b. What is the probability that out of the 6 selected at random, exactly one will be defective?

c. What is the probability that of the 6, two will be defective?

d. What is the probability that of the 6, three will be defective?

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e. What is the probability that of the 6, four will be defective?

f. What is the probability that of the 6, five will be defective?

g. What is the probability that of the 6, all will be defective?

h. With the information from a-g, graph a binomial probability distribution.

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*i. What is the probability that less than 3 will be defective?

*j. What is the probability that at least 3 will be defective?

*k. What is the probability that more than 3 will be defective?

2. Eighty percent of employees use direct deposit. Suppose we select a random sample of 7 employees and count the number using direct deposit. What is the probability that:

a. none use direct deposit?

b. exactly 3 uses direct deposit?

c. exactly 4 uses direct deposit?

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3. The postal service reports ninety percent of its first class mail delivered within 2 days. If 5 letters are randomly selected, what is the probability that:

a. exactly one arrives within 2 days?

b. exactly 3 arrives within 2 days?

c. all arrives within 2 days?

4.5 The mean and standard deviation of the binomial distribution

4.5.1/4.5.2 Calculate the mean and standard deviation of a binomial distribution.

The mean of a probability distribution is often called the expected value of the distribution.  This terminology reflects the idea that the mean represents a "central point" or "cluster point" for the entire distribution.

The mean and variance of a binomial distribution can be computed by these formulas:

Examples:

Calculate the mean, variance, and standard deviation for the binomial distribution problems #1-3 above.

1.

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2.

3.

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Worksheet for 4.0

1. Compute the mean and variance of the following discrete probability distribution.

x P(x)0 0.21 0.42 0.33 0.1

2. Compute the mean and variance of the following discrete probability distribution.

x P(x)2 0.58 0.310 0.2

3. Which of the following variables are discrete and which are continuous random variables?

a. The number of new accounts established by a salesperson in a year.b. The time between customer arrivals to a bank ATM.c. The number of customers in Big Nick’s barber shop.d. The amount of fuel in your car’s gas tank last week.e. The number of minorities on a jury.f. The outside temperature today.

4. Dan Woodward is the owner and manager of Dan’s Truck Stop. Dan offers free refills on all coffee orders. He gathered the following information on coffee refills. Compute the mean, variance, and standard deviation for the distribution of number of refills.

x Percent0 301 40

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2 203 10

5. The director of admissions at a university in estimated the distribution of student admissions for the fall semester on the basis of past experience. What is the expected number (i.e. what is the mean) of admissions for the fall semester? Compute the variance and the standard deviation of the number of admissions.

admissions

Probability

1000 061200 031500 0.1

6. The following table lists the probability distribution for cash prizes in a lottery conducted at Lawson’s Department Store.

x P(x)0 0.45

10 0.30100 0.20500 0.05

If you buy a single ticket, what is the probability that you win:

a. exactly $100?b. at least $10?c. no more than $100?d. Compute the mean, variance, and standard deviation of this distribution.

7. You are asked to match three songs with the performers who made those songs famous. If you guess, the probability distribution for the number of correct matches is:

X (the number correct)

P(x)

0 0.3331 0.5002 0

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3 0.167

What is the probability that you get:

a. exactly one correct?b. at least one correct?c. exactly two correct?d. Compute the mean, variance, and standard deviation of this distribution.

8. In a binomial situation n=4 and π=0.25. Determine the probabilities of the following events using the binomial formula.

a. x=2b. x=3

9. In a binomial situation n=5 and π=0.40. Determine the probabilities of the following events using the binomial formula.

a. x=1b. x=2

10. Assume a binomial distribution where n=3 and π=0.60.

a. Refer to Appendix A, and list the probabilities for values of x from 0 to 3.b. Determine the mean and standard deviation of the distribution.

11. Assume a binomial distribution where n=5 and π=0.30.

a. Refer to Appendix A, and list the probabilities for values of x from 0 to 5.b. Determine the mean and standard deviation of the distribution.

12. A society of investors’ survey found 30 percent of individual investors have used a discount broker. In a random sample of nine individuals, what is the probability:a. exactly two of the samples individuals have used a discount broker?b. exactly four of the them have used a discount broker?c. none of them have used a discount broker?

13. The postal service reports 95 percent of first class mail within the same city is delivered within two days of the time of mailing. Six letters are randomly sent to different locations.a. What is the probability that all sic arrive within two days?

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b. What is the probability that exactly five arrive within two days?c. Find the mean number of letters that will arrive within two days.d. Compute the variance and standard deviation of the number that will arrive within two days.

14. The industry standards suggest that 10 percent of new vehicles require warranty service within the first year. Jones sold 12 Nissans yesterday.a. What is the probability that none of these vehicles requires warranty service?b. What is the probability that exactly one of these vehicles requires warranty service?c. Determine the probability that exactly two of these vehicles require warranty service.d. Compute the mean and standard deviation of this probability distribution.