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The first card selected from a deck of cards was a king

If it is returned to the deck, what is the probability that a king will be drawn on the second selection?

If it is not replaced, what is the probability that a king will be drawn on the second selection?

What is the probability that a king will be selected on the first draw from a deck and another king on the second draw (assuming the first king was not replaced)?

Mr. and Mrs. Wilhelms are both retired and living in a retirement community in Arizona. Suppose the probability that a retired man will live another 10 years is .60. The probability that a retired woman will live another 10 years is .70.

What is the probability that both will be alive 10 years from now?

What is the probability that in 10 years Mr. Wilhelms is not living and Mrs. Wilhelms is living?

What is the probability that in 10 years at least one is living?

There are 400 employees at G. G. Greene Manufacturing Co. and 100 of them smoke. there are 250 men working for the company, and 75 of them smoke. What is the probability that an employee selected at random:

Is a man?

Smokes?

Is male and smokes?

Is male or smokes?

Flashner Marketing Research, Inc. Specializes in providing assessments of the prospects for women’s apparel shops in shopping malls. Al Flashner, president, reports that he assesses the prospects as good, fair, or poor

Records from previous assessments show that 60 % of the time the prospects were rated as good, 30 % of the time fair, and 10 % of the time poor

Of those rated good, 80 % made a profit the first year; of those rated fair, 60 percent made a profit the first year; and of those rated poor; 20 % made a profit the first year.

Connie’s Apparel was one of Flashner’s clients and made a profit in their first year.

What is the probability that it was given an original rating of poor?

A random variable is a numerical value determined by the outcome of an experiment.Random Variables

A probability distribution is the listing of all possible outcomes of an experiment and the corresponding probability.

A discrete probability distributioncan assume only certain outcomes.Types of Probability Distributions

- A continuous probability distributioncan assume an infinite number of values within a given range.

Examples of a discrete distribution are:

The number of students in a class.

The number of children in a family.

The number of cars entering a carwash in a hour.

Number of home mortgages approved by Coastal Federal Bank last week.

Types of Probability DistributionsExamples of a continuous distribution include:

The distance students travel to class.

The time it takes an executive to drive to work.

The length of an afternoon nap.

The length of time of a particular phone call.

Types of Probability DistributionsThe main features of a discrete probability distribution are:

The sum of the probabilities of the various outcomes is 1.00.

The probability of a particular outcome is between 0 and 1.00.

The outcomes are mutually exclusive.

Features of a Discrete DistributionEXAMPLE 1: Consider a random experiment in which a coin is tossed three times. Let x be the number of heads. Let H represent the outcome of a head and T the outcome of a tail.Example 1

The possible outcomes for such an experiment will be:

TTT, TTH, THT, THH,

HTT, HTH, HHT, HHH.

EXAMPLE 1 continued- Thus the possible values of x (number of heads) are 0,1,2,3.

The outcome of zero heads occurred once.

The outcome of one head occurred three times.

The outcome of two heads occurred three times.

The outcome of three heads occurred once.

From the definition of a random variable, x as defined in this experiment, is a random variable.

EXAMPLE 1 continuedThe mean:

reports the central location of the data.

is the long-run average value of the random variable.

is also referred to as its expected value, E(X), in a probability distribution.

is a weighted average.

The Mean of a Discrete Probability DistributionThe mean is computed by the formula:

where represents the mean and P(x) is the probability of the various outcomes x.

The Mean of a Discrete Probability DistributionThe variance measures the amount of spread (variation) of a distribution.

The variance of a discrete distribution is denoted by the Greek letter 2 (sigma squared).

The standard deviation is the square root of 2.

The Variance of a Discrete Probability DistributionThe variance of a discrete probability distribution is computed from the formula:The Variance of a Discrete Probability Distribution

Dan Desch, owner of College Painters, studied his records for the past 20 weeks and reports the following number of houses painted per week:EXAMPLE 2

Compute the mean and variance:

Compute the mean and variance:

a. Which is a real probability distribution?

b. Find the probability that x is:

3. More than 5

1. Exactly 15

2. No more than 10

Compute the variance and standard deviation:

Find the probability that you get:

c. Exactly 2

a. Exactly 1

b. At least 1

d. Compute the mean, variance and standard deviation

The binomial distribution has the following characteristics:Binomial Probability Distribution

- An outcome of an experiment is classified into one of two mutually exclusive categories, such as a success or failure.
- The data collected are the results of counts.
- The probability of success stays the same for each trial.
- The trials are independent.

To construct a binomial distribution, let

n be the number of trials

x be the number of observed successes

be the probability of success on each trial

- The formula for the binomial probability distribution is:

The Alabama Department of Labor reports that 20% of the workforce in Mobile is unemployed. From a sample of 14 workers, calculate the following probabilities:

Exactly three are unemployed.

At least three are unemployed.

At least none are unemployed.

- The probability of exactly 3:

- The probability of at least 3 is:

From Alabama, recall that =.2 and n=14.

Hence, the meanis:

=n = 14(.2) = 2.8.

The varianceis:

2= n(1- ) = (14)(.2)(.8) =2.24.

The US Postal Service reports that 95% of first class mail within the same city is delivered within 2 days of the time of mailing. Six letters are randomly sent to different locations.

a. What is the probability that all 6 arrive within 2 days

b. What is the probability that exactly 5 arrive within 2 days

c. Find the mean number of letters that will arrive within 2 days

d. Compute the variance and standard deviation of the number that will arrive within 2 days.

A finite population is a population consisting of a fixed number of known individuals, objects, or measurements. Examples include:Finite Population

- The number of students in this class.
- The number of cars in the parking lot.
- The number of homes built in Blackmoor.

For use when the probability does not stay the same

The hypergeometric distribution has the following characteristics:

There are only 2 possible outcomes.

The probability of a success is not the same on each trial.

It results from a count of the number of successes in a fixed number of trials.

Hypergeometric DistributionThe formula for finding a probability using the hypergeometric distribution is:

where Nis the size of the population,

S is the number of successes in the population,

xis the number of successes in a sample of

n observations.

Hypergeometric DistributionUse the hypergeometric distribution to find the probability of a specified number of successes or failures if:

the sample is selected from a finite population without replacement (recall that a criteria for the binomial distribution is that the probability of success remains the same from trial to trial).

Hypergeometric Distribution- the size of the sample n is greater than 5% of the size of the population N .

The National Air Safety Board has a list of 10 reported safety violations. Suppose only 4 of the reported violations are actual violations and the Safety Board will only be able to investigate five of the violations. What is the probability that three of five violations randomly selected to be investigated are actually violations?

When the probability of success is small and the population is very large

The binomial distribution becomes more skewed to the right (positive) as the probability of success become smaller.

Poisson Probability Distribution- The limiting form of the binomial distribution where the probability of success is small and n is large is called the Poisson probability distribution.

Examples:

defective parts in a shipment waiting to go out

The binomial distribution becomes more skewed to the right (positive) as the probability of success become smaller.

Poisson Probability Distribution- The limiting form of the binomial distribution where the probability of success is small and n is large is called the Poisson probability distribution.

ThePoissondistribution can be described mathematically using the formula:

where is the mean number of successes in a particular interval of time, e is the constant 2.71828, andx is the number of successes.

Poisson Probability DistributionThe mean number of successes can be determined in binomial situations by n, where n is the number of trials and the probability of a success.Poisson Probability Distribution

- The variance of the Poisson distribution is also equal to n.

The Sylvania Urgent Care facility specializes in caring for minor injuries, colds, and flu. For the evening hours of 6-10 PM the mean number of arrivals is 4.0 per hour. What is the probability of 4 arrivals in an hour?

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