Chapter 5: Probability Chapter 5: Probability DistributionsDistributions

Random VariablesRandom Variables

• A random variablerandom variable is a quantitative variable whose values are determined by chance.

• DiscreteDiscrete random variables have values that can be countedcounted.

• Continuous randomContinuous random variables are obtained from data that can be measuredmeasured rather than countedcounted.

ExampleExample• Example: Which of the following

random variables are discrete and which are continuous?

• Measure the time it takes a student selected at random to register for the summer term?

• Count the number of bad checks drawn on a local bank on a day selected at random.

Example (cont.)Example (cont.)

• Example: Which of the following random variables are discrete and which are continuous?

• Measure the amount of gas needed to drive your car 200 miles.

• Pick a random sample of 50 registered voters in a district and find the number who voted in the last county election.

Discrete Probability Discrete Probability DistributionDistribution

• A discrete probability discrete probability distributiondistribution consists of the valuesvalues a random variable can assume and the corresponding probabilitiescorresponding probabilities of the values. The probabilities are determined theoreticallytheoretically or by observationobservation.

Example: Boredom Tolerance Example: Boredom Tolerance TestTest

• Dr. Fidget developed a test to measure boredom tolerance. He administered it to a group of 20,000 adults between the ages of 25 and 65. The possible scores were 0, 1, 2, 3, 4, 5 and 6, with 6 indicating the highest tolerance for boredom. The results were as follows. Find the probability distribution for the scores and illustrate the distribution.

Boredom Tolerance Test

Scores for 20,000 Subjects

Score Number of

Subjects

0 1,400

1 2,600

2 3,600

3 6,000

4 4,400

5 1,600

6 400

Example: Boredom Tolerance Example: Boredom Tolerance TestTest

Boredom Tolerance Test

Scores for 20,000 Subjects

Score Number of

Subjects

0 1,400

1 2,600

2 3,600

3 6,000

4 4,400

5 1,600

6 400

ExampleExample

• Determine the probability distribution for the number of heads in a toss of three coins.

Two Requirements for a Two Requirements for a Probability DistributionProbability Distribution

• The sumsum of the probabilities of all the events in the sample space must equal 1; that is ________________

• The probability of each event in the sample space must be betweenbetween or equalequal to 0 or 11. That is, _______________.

Example:Example:• Determine whether the distribution

represents a probability distribution.

X 3 6 8

P(x) -0.3 0.6 .7

X 20 30 40 50

P(x) 1.1 0.2 0.9 0.3

ExampleExample

• Determine the probability distribution for the sum of two randomly tossed dice.

Mean & Variance of a Mean & Variance of a Probability DistributionProbability Distribution

• The mean of a random variable with a discrete probability distribution is

____________________________

• The variance of a probability distribution is

__________________

ExampleExample

• The probability distribution for the number of customers at the Sunrise Coffee Shop is shown here. Find the mean, variance and standard deviation of the distribution.

Number of Customers, X

50 51 52 53 54

P(x) 0.10 0.2 0.37 0.21 0.12

Expected ValueExpected Value

• The expected valueexpected value of a discrete random variable of a probability distribution is the theoretical theoretical averageaverage of the variable. The formula is = E(X) = ___________. The symbol E(X) is used for the expected value.

• Example: If a 60-year-old buys a $1000 life insurance policy at a cost of $60 and has a probability of 0.972 of living to age 61, find the expectation of the policy.

ExampleExample

• At a carnival you pay $2.00 to play a coin-flipping game with three fair coins. On each coin on one side has the number 0 and the other side has the number 1. You flip the three coins at one time and you win $1 for every 1 that appears on top. Are your expected earnings equal to the cost to play?

Binomial DistributionBinomial Distribution

• A binomial experimentbinomial experiment is a probability experiment that satisfies the following four requirements:

• Each trial can have two outcomes or outcomes that can be reduced to two outcomes. These outcomes can be considered as either success or failure

• There must be a fixed number of trials.• The outcomes of each trial must be

independent of each other.• The probability of a success must remain

the same from trial to trial.

Notation for the Binomial Notation for the Binomial DistributionDistribution

• P(S) The symbol for the probability of success

• P(F) The symbol for the probability of failure

• p The numerical probability of a success

• q The numerical probability of a failure

__________________ __________________

• n The number of trials• X The number of successes in n trials

Binomial Probability Binomial Probability FormulaFormula

• In a binomial experiment, the probability of exactly X successes in n trials is

• _____________________________

Example: Hybrid TomatoExample: Hybrid Tomato

• A biologist is studying a new hybrid tomato. It is known that the seeds of this hybrid tomato have probability 0.70 of germinating. The biologist plants 10 seeds. What is the probability that exactly 8 seeds will germinate?

Example: Blood Type BExample: Blood Type B

• According to the Textbook of Medical Physiology, 5th edition, by Arthur Guyton, 9% of the population has blood type B. Suppose we choose 18 people at random from the population and test the blood type of each. What is the probability that three of these people have blood type B?

• Example: A burglar alarm system has 6 fail-safe components. The probability of each failing is 0.05. Find these probabilities.

• Exactly 3 will fail

• Fewer than 2 will fail

• None will fail

• The mean, variance and standard deviation of a variable that has the binomial distribution can be found by using the following formulas.

• Mean _________________• Variance _________________• Standard

deviation ________________

Example: Providence Example: Providence ElectronicsElectronics

• Providence Electronics receives 400 calls in a typical day. Seventy percent of callers have touch-tone phones. If x is the random variable representing the number of callers with touch-tone phones in a group of 400, find its mean and standard deviation.

ExampleExample

• : In a restaurant, a study found that 42% of all patrons smoked. If the seating capacity of the restaurant is 80 people, find the mean, variance, and standard deviation of the number of smokers. About how many seats should be available for smoking customers?