Chapter 5 Discrete Probability Distributions

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Chapter 5 Discrete Probability Distributions

• Mrs. Mandy Wimpey
• Palmetto High School

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Real-World Application

• DIAGNOSTIC TESTING
• When a disease is rare, health care workers
  will pool the blood samples of many people and
  test the combination of blood. This saves time
  and money.
• If the pooled sample results in a negative test,
  no further testing is needed (no one has the
  disease).
• If the pooled sample results in a positive test,
  further testing will be conducted on an
  individual basis (since at least one person has
  the disease).
• Lets suppose the probability of having a disease
  is 0.05 and a pooled sample of 15 individuals is
  tested. What is the probability that no further
  testing will be needed?

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Introduction and Terms

• Random Variable a variable whose values are
  determined by chance
• Discrete Random Variable a variable whose
  values are found by counting
• Discrete Probability Distribution the values a
  random variable can assume and their
  corresponding probabilities.

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Constructing a Discrete Probability Distribution
Construct a probability distribution for rolling
one die. List all possible outcomes on the
first row. These are all your x-values. List
each x-values chance of occurring on the second
row. These are the corresponding
probabilities for each x-value.
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Constructing a Discrete Probability Distribution

• A coin is tossed 3 times. Construct a
  probability distribution for the number of heads
  you get in 3 tosses.

Recall the sample space for a coin tossed 3
times. Do a tree diagram if you have forgotten.
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Requirements for a Probability Distribution

• The total (sum) of the probabilities must equal
  1.0.
• Each individual probability must be between 0 and
  1, inclusively.

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Mean, Standard Deviation, and Expected Value

• The mean (average) of a probability distribution
  is calculated as
• The standard deviation of a probability
  distribution is calculated as
• The expected value of a probability distribution
  is the mean of the probability distribution.

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Steps in calculating the Mean and Standard
Deviation

• The Mean
• Multiply each x by its corresponding P(x).
• Add them up.
• This is the mean or expected value for the
  distribution.
• The Standard Deviation
• Square each x value.
• Multiply each squared x by its corresponding
  P(x) value.
• Add them up.
• Subtract the mean from the sum you found in step
  3.
• Take the square root of the difference you found
  in step 4.

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Using the TI-84 to calculate the Mean Standard
Deviation for a Probability Distribution

• Enter your x values into L1.
• Enter your P(x) values into L2.
• Press STAT
• Go over to CALC
• Select 1-VAR STATS
• Type L1, L2 to tell the calculator to use both
  lists at one time
• Press ENTER.

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Examples

• Example 1
• The probability that 0, 1, 2, 3, or 4 people will
  be placed on hold when they call a radio talk
  show is shown in the distribution.
• Find the expected number of people on hold and
  the standard deviation.
• Should the radio station get an additional phone
  line? Explain.

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The expected number of people on hold is the mean
number of people on hold. This is calculated by
The radio station can expect an average of 1.6
people to be on hold at any given time.
The standard deviation of the number of people on
hold is found as follows
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More Expected Value

• Example One thousand tickets are sold at 1
  each for a color TV valued at 350. What is the
  expected value of the gain if a person purchases
  one ticket?
• The problem must be set up using a table as
  follows in the next slide.

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If you win, you have gained only 349 since you
paid a dollar to enter the contest. Since there
are 1000 tickets and 1 winning ticket, you have a
1/1000 chance of winning. If you lose, you lost
1, so its a negative amount. Since there is
only 1 winning ticket, there are 999 that are
non-winning tickets.
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• To calculate the Expected Value, you are
  calculating the Mean. Multiply each x by its
  corresponding P(x) value. Then, add.
• E(x) (349)(1/1000) (-1)(999/1000)
• -0.65
• One individual person either wins a 350 TV or
  loses 1. So the -0.65 concerns the average
  loss over a period of time.

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The Binomial Probability Distribution

• Requirements for a Binomial Distribution
• Each trial can have only 2 outcomes. We call
  these successes and failures.
• There is a fixed number of trials. (The
  experiment doesnt go on forever.)
• Each trial is independent.
• The probability of a success remains constant for
  each trial.

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Notation for Binomial Distributions

• P(S) probability of a success
• P(F) probability of a failure
• p numerical probability of success
• q numerical probability of failure
• P(S) p and P(F) q
• n total number of trials
• x number of successes in n trials
• q 1- p

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The Binomial Probability Formula
Using the TI-84 will help simplify this formula.
You will use the MATH menu. The formula will
then look like
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Examples

• Example 1
• A survey found that one out of five Americans say
  he or she has visited a doctor in any given
  month. If 10 people are selected at random, find
  the probability that exactly 3 will have visited
  a doctor last month.
• In this case, n 10 x 3 p 1/5 and q 4/5.
• So, the formula should look like this

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• Example 2
• A survey found that 30 of teenage consumers
  receive their spending money from part-time jobs.
  If 5 teenagers are randomly selected, find the
  probability that at least 3 of them have
  part-time jobs.
• For this problem, n 5 x 3, 4, 5 p 0.30
  and q 0.70
• Putting it all together in the formula, we get

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Mean and Standard Deviation for a Binomial
Distribution

• Mean
• Standard Deviation

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Example

• Example 1
• A coin is tossed 4 times. Find the mean and
  standard deviation of the number of heads
  obtained.
• Mean
• Standard Deviation

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