Section 5.2

Purpose of Section Purpose of Section Purpose of Section Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. complete ordered field. complete ordered field. complete ordered field. The axioms which describe the arithmetic of the real numbers form an algebraic field algebraic field algebraic field algebraic field. The order axioms combined with the field axioms form a mathematical structure known as an ordered field ordered field ordered field ordered field, , , , and finally the completeness axiom completeness axiom completeness axiom completeness axiom are combined with the ordered field axioms to give us the real number system.

• Distinguish between discrete random variables and continuous random variables • Construct a discrete probability distribution and its graph • Determine if a distribution is a probability distribution • Find the mean, variance, and standard deviation of a discrete probability distribution • Find the expected value of a discrete probability distribution

Discrete Probability Distributions
Discrete probability distribution • Lists each possible value the random variable can assume, together with its probability. • Must satisfy the following conditions:

In Words
In Symbols 1. The probability of each value of the discrete random variable is between 0 and 1, inclusive.

Example: Constructing a Discrete Probability Distribution
An industrial psychologist administered a personality inventory test for passive-aggressive traits to 150 employees. Individuals were given a score from 1 to 5, where 1 was extremely passive and 5 extremely aggressive. A score of 3 indicated neither trait. Construct a probability distribution for the random variable x. Then graph the distribution using a histogram.
• Divide the frequency of each score by the total number of individuals in the study to find the probability for each value of the random variable.

Probability, P(x)
Score, x

Passive-Aggressive Traits
Because the width of each bar is one, the area of each bar is equal to the probability of a particular outcome.

Example: Finding the Mean
The probability distribution for the personality inventory test for passive-aggressive traits is given. Find the mean score. Solution:

Variance and Standard Deviation
Variance of a discrete probability distribution

Example: Finding the Variance and Standard Deviation
The probability distribution for the personality inventory test for passive-aggressive traits is given.

Solution: Finding the Variance and Standard Deviation
Recall μ = 2.94

Expected Value
Expected value of a discrete random variable • Equal to the mean of the random variable.
• To find the gain for each prize, subtract the price of the ticket from the prize:  Your gain for the $500 prize is $500 -$2 = $498  Your gain for the $250 prize is $250 -$2 = $248  Your gain for the $150 prize is $150 -$2 = $148  Your gain for the $75 prize is $75 -$2 = $73 • If you do not win a prize, your gain is $0 -$2 = -$2 You can expect to lose an average of $1.35 for each ticket you buy.
• Distinguished between discrete random variables and continuous random variables • Constructed a discrete probability distribution and its graph • Determined if a distribution is a probability distribution • Found the mean, variance, and standard deviation of a discrete probability distribution • Found the expected value of a discrete probability distribution

Section 5.3 & 5.4 Objectives
• Determine if a probability experiment is a binomial experiment • Find binomial probabilities using the binomial probability formula • Find binomial probabilities using technology, formulas, and a binomial probability table • Graph a binomial distribution • Find the mean, variance, and standard deviation of a binomial probability distribution

Binomial Experiments
1. The experiment is repeated for a fixed number of trials, where each trial is independent of other trials.
2. There are only two possible outcomes of interest for each trial. The outcomes can be classified as a success (S) or as a failure (F).

Symbol Description n
The number of times a trial is repeated

p = P(S)
The probability of success in a single trial q = P(F) The probability of failure in a single trial (q = 1 -p) x The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, … , n.
1. A certain surgical procedure has an 85% chance of success. A doctor performs the procedure on eight patients. The random variable represents the number of successful surgeries.

Solution: Binomial Experiments
Binomial Experiment

Example: Binomial Experiments
Decide whether the experiment is a binomial experiment. If it is, specify the values of n, p, and q, and list the possible values of the random variable x.
2. A jar contains five red marbles, nine blue marbles, and six green marbles. You randomly select three marbles from the jar, without replacement. The random variable represents the number of red marbles.

Not a Binomial Experiment
• The probability of selecting a red marble on the first trial is 5/20.
• Because the marble is not replaced, the probability of success (red) for subsequent trials is no longer 5/20.
• The trials are not independent and the probability of a success is not the same for each trial.

Binomial Probability Formula
Binomial Probability Formula • The probability of exactly x successes in n trials is

number of successes in n trials
Microfracture knee surgery has a 75% chance of success on patients with degenerative knees. The surgery is performed on three patients. Find the probability of the surgery being successful on exactly two patients.

Example: Constructing a Binomial Distribution
In a survey, U.S. adults were asked to give reasons why they liked texting on their cellular phones. Seven adults who participated in the survey are randomly selected and asked whether they like texting because it is quicker than calling. Create a binomial probability distribution for the number of adults who respond yes.
• Find probabilities using the geometric distribution • Find probabilities using the Poisson distribution