CHAPTER 5 Discrete Probability Distribution Objectives

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Construct a probability distribution for a random variable. Find the mean, variance, and expected value for a discrete random variable. Find the exact probability for X successes in n trials of a binomial experiment. Find the mean, variance, and standard deviation for the variable of a binomial distribution.

Introduction • Many decisions in business, insurance, and other real-life situations are made by assigning probabilities to all possible outcomes pertaining to the situation and then evaluating the results. • This chapter explains the concepts and applications of what is called a probability distribution. In addition, special probability distributions, the binomial distribution, is explained. 5.1 Probability Distribution I. Random Variables A random variable is a variable whose values are determined by chance.

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A random variable is discrete if it can potentially assume only a finite or countable number of values.

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A random variable is continuous if it potentially can take on any value on an interval.

Example 1: Identify the following random variables as discrete or continuous. a) Weight of a package b) Number of students in a first-grade classroom c) Age of a cancer patient

Example 2: An experiment consists of tossing five fair coins. Let x be the random variable that is the number of heads in the five tosses. Is x discrete or continuous? List the sample space of values for x.

Example 3: Let x be the random variable that gives the amount of time it takes a person to drive to work. Is x discrete or continuous? List the sample space of values of x.

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II. Discrete Probability Distribution

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A discrete probability distribution consists of the values a random variable can assume and the corresponding probabilities of the values. The probabilities are determined theoretically or by observation.

Example 1: The probabilities that a customer will purchase 0, 1, 2, or 3 books are 0.45, 0.30, 0.15, and 0.10, respectively. (a) Construct a probability distribution for the data. (b) Draw a graph for the distribution.

Example 2: Consider families with three children. Let x be the number of girls in a family. Find the probability distribution for x and construct a graph for the probability distribution.

III. Requirements for a Discrete Probability Distribution 1. P(x) will always be a number between 0 and 1 inclusive: 0 ≤ P(x) ≤ 1 2. The sum of the values of P(x) for each distinct value of x is 1:

∑

P(x) = 1

2

Example 1: A random variable x has this probability distribution: x

0 .2

P(x)

1 .3

2 .1

3 ?

(a) Find P(x = 3)

(b) What is the probability that x is greater than 0?

(c) What value of x is most likely to occur?

(d) What is P(x = 8) ?

Example 2: Determine whether the distribution represents a probability distribution. If it does not, state why. x P(x)

5 1.2

10 0.3

15 0.5

5.2 Mean, Variance, Standard Deviation, and Expectation I. The mean of a discrete probability distribution Suppose two coins are tossed repeatedly, and the number of heads that occurred is recorded. What will be the mean of the number of heads?

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In order to find the mean for a probability distribution, one must multiply each possible outcome by its corresponding probability and find the sum of the products.

= µ

∑ x ⋅ P( x) where X1, X2, X3, …,Xn are the outcomes and P(X1), P(X2), P(X3),…P(Xn) are the corresponding probabilities.

II. Variance of a Probability Distribution

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The variance of a probability distribution is found by multiplying the square of each outcome by its corresponding probability, summing those products, and subtracting the square of the mean.

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The formula for calculating the variance is: σ 2 =

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The formula for the standard deviation is: σ = σ 2

∑ (x − µ)

2

⋅ P( x)

Example 1: For the probability distribution given below, find (a) the mean, (b) the variance and (c) the standard deviation. Also (d) construct a graph for the probability distribution and describe the shape of the distribution. x P(x) ________________________________________________________________________ 0

1/10

1

4/10

2

3/10

3

2/10

a) The mean b) Variance

c) Standard deviation d) Construct a graph for the probability distribution and describe the shape of the distribution.

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Example 2: Use the computational formula to find the standard deviation of the probability distribution given in example 1.

III. Expected Value

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Expected value or expectation is used in various types of games of chance, in insurance, and in other areas, such as decision theory. The expected value of a discrete random variable of a probability distribution is the theoretical average of the variable. The formula is: E ( x= x ⋅ P( x) ) µ=

∑

For a discrete random variable x, the expected value of x is the mean of the random variable x. The symbol E(X) is used for the expected value.

Example 1: (Ref: General Statistics by Chase/Bown, 4th Ed.) A high school class decides to raise some money by conducting a raffle. The students plan to sell 2000 tickets at $1 apiece. They will give one prize of $100, two prizes of $50, and three prizes of $25. If you plan to buy one ticket, what are your expected net winnings?

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Example 2: (Ref: Elementary Statistics by Triola, 9th edition) In New Jersey’s Pick 4 lottery game, you pay 50 cents to select a sequence of four digits, such as 7273. If you win by selecting the same sequence of four digits that are drawn, you collect $2,788. (a) How many different selections are possible?

(b) What is the probability of winning?

(c) If you win, what is your net profit?

(d) Find the expected value.

5.3 The Binomial Distribution I. Binomial Experiment • Many types of probability problems have only two possible outcomes or they can be reduced to two outcomes. • Examples include: when a coin is tossed it can land on heads or tails, when a baby is born it is either a boy or girl, etc The Binomial Experiment

The binomial experiment is a probability experiment that satisfies these requirements: 1. Each trial can have only two possible outcomes—success or failure. 2. There must be a fixed number of trials. 3. The outcomes of each trial must be independent of each other. 4. The probability of success must remain the same for each trial. The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution 6

Example 1: (Ref: Exploring Statistics by Kitchens, 2nd ed.) Which of the following are binomial random variables? (a) The number of successful heart transplants out of five patients.

(b) The length of a prison term for possession of marijuana

(c) The name of each student in Math 227

(d) The number of approved food stamp recipients out of 50 applications

II. Notation for the Binomial Distribution • The symbol for the probability of success • The symbol for the probability of failure • The numerical probability of success • The numerical probability of failure • The number of trials • The number of successes III. Binomial Probability Formula A binomial experiment consists of n identical trials with probability of success p on each trial. The probability of x successes in n trials is

= P( x)

n ⋅ p x ⋅ q n− x (n − x)! x !

for = x 0,1, 2,....., n

OR

P( x) =

n

Cx ⋅ p x ⋅ q n − x

for x = 0,1, 2,....., n

Example 1: (a) Find

8C3

(b) Find

12C7

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Example 2: Consider a binomial experiment with n = 15, p = .3, and x = number of success. Use the Binomial Formula for P(x) to find the probability that (a) x = 11

(b) x is less than 2

Example 3: It was found that 68% of American victims of health care fraud are senior citizens. If 10 victims are randomly selected, find the probability that exactly 3 are senior citizens.

Example 4: Consider a binomial experiment with n = 12, p = .4, and x = number of success. Use the Binomial Table to find the probability that (a) x is greater than 5 but less than 8

(b) x is greater than 7

(c) x equals 6

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IV. Mean and Variance of a Binomial Distribution For a binomial experiment consisting of n trials with the probability of success p, the mean or expected of x is

For a binomial experiment consisting of n trials with the probability of success p, the variance of x is

The standard deviation of x is

Example 1: Assume that 60% of a college’s student loan applications are approved. Ten applications are chosen at random. (a) What is the probability that eight or more are approved?

(b)

How many applications are expected to be approved? (Find the mean of the number approved out of ten applications.)

(c) What is the standard deviation of the number approved out of ten applications?

Summary • A probability distribution can be graphed, and the mean, variance, and standard deviation can be found. • The mathematical expectation can also be calculated for a probability distribution. • Expectation is used in insurance and games of chance. • The binomial distribution is used when there are only two outcomes for an experiment, a fixed number of trials, the probability is the same for each trial, and the outcomes are independent of each other. Conclusion

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Many decisions in business, insurance, and other real-life situations are made by assigning probabilities to all possible outcomes pertaining to the situation and then evaluating the results. 9