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Chapter 5: Discrete Probability Distributions (PD)
5-2
Probability Distribution: P(X)
It’s a graph, table, or formula that gives the ______________ for each value of random variable (In
the case of DRV, see below: It is a list of D.R.V. and their probabilities.)
Requirements:
1)
2)
3)
Random Variable (X)
It is a single numerical value for each outcome of a procedure.
Discrete Random variable (D.R.V.)
It is a variable which may take on only a ___________ number of distinct values such as
0,1,2,3,4,........ Discrete random variables are usually counts. Examples of discrete random variables
include the number of children in a family, the number of patients in a doctor's surgery, and the
number of defective light bulbs in a box of ten.
Continuous R.V
It has _______________ many values and those values can be associated with measurements on a
continuous scale without gaps.
Example 1:
Discrete or Continuous?
Random Variable
the number of heads when three coins are tossed
the speed of a race car
the number of people who play the lottery each day
the number of births at a hospital on a specific day
the weight of an elephant
the highest temperature recorded on a specific day
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Example 2: Determine if the
following is a probability distribution.
x
0
1
2
3
4
5
P (x )
0.243
0.167
0.213
0.149
0.232
0.164
Example 3: Determine the required
value of the missing probability to
make the distribution a discrete
probability distribution.
x
P(x)
0
0.25
1
0.30
2
?
3
0.10
Example 4: A coin is tossed three times. Suppose that the random variable X is defined as the
number of heads that appear when a coin is tossed three times.
a.
Is this an example of a Discrete or Continuous Random Variable? What are the possible
values for X? List them under “X” in the table below.
b.
Write the sample space in some organized manner.
c.
Let Random Variable X be the number of heads in each outcome. Count the number of
heads in each outcome in b) and complete the probability distribution table.
Random Variable x = the number of heads
X
P(X)
P(x)
X
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A probability histogram is a histogram in which the horizontal axis corresponds to the value of the
random variable and the vertical axis represents the probability of that value of the random variable.
x
P(x)
0
0.06
1
0.58
2
0.22
3
0.10
4
0.03
5
0.01
Mean (expected value) & Standard Deviation for a probability distribution
Use same method as you do for grouped data, let x column be L1 and P(x) be L2
Calculate: 1-Var Stats L1,L2
Mean (expected value):
μ = what shows up as x
formula μ =
Standard Deviation: (note use of parameter symbols)
σ = what shows up as σ x
n
∑ x ⋅ P( x )
i =0
i
i
formula: (see book)
Rounding rule for μ, σ, σ2 : one more decimal place than # of places used for
random variable x (i.e. column 1)
Example 4 Continue: Find the ACTUAL μ
Using the formula above, multiply each possible outcome with it’s corresponding probability and sum:
n
1
3
3
1
μ = ∑ xi ⋅ P( xi ) = 0 ⋅ + 1 ⋅ + 2 ⋅ + 3 ⋅ = 1.5
8
8
8
8
i =0
We know we cannot toss three coins and end up with 1.5 heads. What this means is that if we do
this say… forever, in the long run, we would “expect” to have an average of _____ heads. Therefore,
our expected value is _____.
** Now use your calculator to find μ = _______ & σ = _______
X
P(x)
0
0.15
1
0.45
2
0.30
3
0.10
Example 5: x represents the number of computers in a household. Find the mean and the
standard deviation.
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Class Activity A – in your CA book –
A ski resort loses $70,000 per season when it does not snow very much and makes $250,000 profit when
it does snow a lot. The probability that it will snow at least 75” (ie. A good season) is 40%.
1.
Using the following table to guide you, find the expectation for the profit and explain what it
means.
Expected value = ___________, means ……
X
Good Season
Bad Season
P(X)
2. What would it mean if the expected value was 0?
3. What would it mean to the ski resort if the expected value
was negative?
TWO Methods for Identifying Unusual Results: (Note: Unusual ≠ outlier)
1) Using Range Rule of thumb is SAME as before:
Value is usual if it is within ____ standard deviations of the mean
2) Using probabilities
 Unusually high: x successes among n trials is an unusually high number of successes
if P(x or more) ≤ 0.05.
 Unusually low: x successes among n trials is an unusually low number of successes if
P(x or fewer) ≤ 0.05.
Example 6: Based on information from MRINetwork, some job applicants are required to have
several interviews before a decision is made. The number of required interviews and the
corresponding probabilities are: 1 (0.09); 2 (0.31); 3(0.37); 4(0.12); 5(0.05); 6 (0.06).
a) Does the given information describe a probability distribution?
b) Assuming that a probability distribution is described, find its mean and standard deviation.
(Use a graphing calculator)
c) Use the range of thumb to identify the range of values for usual numbers of interviews.
d) Is it unusual to have a decision after one interview? Explain.
e) Are any of the outcomes unusually high or low?
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5 – 3 Binomial Probability Distributions **** SPECIAL STUFF!!! ***
Many probability experiments have only two outcomes. Either we have a pop quiz tomorrow or we
don’t, either I get the question right or wrong, either the couple has a girl or a boy. This type of
probability distribution is called a binomial probability distribution.
There must be certain criterion met before we can call an experiment a binomial probability
experiment:
1.
2.
3.
4.
There must be a fixed number of trials.
For each trial there are two mutually exclusive outcomes. Typically these are referred to as
success and failure.
The trials are independent of one another. That is, one does not affect the outcome of the
other.
The probability of success is fixed for each trial of the experiment.
Example 7: Determine if each of the following is an example of a Binomial Experiment. State why or
why not.
a) A player rolls a fair die 10 times. The number, x, of 3’s rolled is recorded.
Success =
Failure =
b) Asking 50 adults the amount of their last cell phone bill
c) Asking 30 homeowners if they would favor a new tax to support education. Assume that the
probability of a randomly selected homeowner favoring a new tax is 0.7.
Notation for Binomial Probability Distributions:
S and F will represent the two possible outcomes (success and failure). p and q will represent their
corresponding probabilities, respectively.
P(S) = p
p = probability of success
P(F) = 1 – p = q
q = probability of failure
n = fixed number of trials
x = number of successes in n trials
p = the probability of success in any one of the trials
q = the probability of failure in any one of the trials
P(x) = probability of getting exactly x successes among the n trials
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A) Finding probability of Exactly ____ # successes occurring in a trial
∙How to find the binomial probabilities using your calculator:
We use this when we want to find the probability of exactly a certain number of successes occurs i.e. P (x = 3)
What you want
↓
=
P ( x #)
=
what you type
↓
binompdf(n,p,x)
(n,p,x defined as given above.)
Press 2nd VARS (to get to DISTR, which stands for “distributions”), then select the option identified as
binompdf, “binomial probability distribution function.” Complete the entry of binompdf(n,p,x)
with the specific values for n, p and x then press ENTER.
Example 8: An experiment consists of flipping a fair coin 8 times and counting the number of tails.
Find the probability of getting exactly 3 tails.
Success =
Failure =
Class Activity B: Forty-three percent of marriages end in divorce. You randomly select 15 married
couples. Find the probability exactly 5 of the marriages will end in divorce.
________________________________________________________________________________
B) What about finding Less than or equal to a certain number of “successes”?
We use this when we want to add a series of probabilities together i.e.
P (x ≤ 2) = P(x = 0) + P(x = 1) + P(x = 2)
P(x ≤ # ) = binomcdf(n,p,x)
Press 2nd VARS (to get to DISTR, which stands for “distributions”), then select the option identified as
binomcdf, “binomial cumulative distribution function” Complete the entry of binomcdf(n,p,x) or
binomcdf(n,p,x) with the specific values for n, p and x then press ENTER.
Example 8: Fifteen peas are generated from parents having the green/yellow pair of genes, so there
is a 0.75 probability that an individual pea will have a green pod. Find the probability that among the
15 offspring peas, no more than 5 has a green pod.
Example 9: Researchers conducted a study and found that the graduation rate for the medical
students admitted through special programs was 91%. If 12 of the students from the special programs
are randomly selected, find the probability that at least 11 of them graduated.
Class Activity C: The Write Right Company manufactures ballpoint pens and has been experiencing
a 5% rate of defective pens. Find the probability that among 50 pens,
a) none are defective.
b) at least one is defective.
c) at most 7 are defective.
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5 – 4 Mean, Variance, and Standard Deviation for the Binomial Distribution
The formulas for the mean, variance and standard deviation of a variable that has the binomial
distribution are as follows.
Mean: μ = np
Standard Deviation: σ =
Variance σ2 = npq
npq
Recall: n = number of trials, p = probability of “success”, q = “probability of “failure”
Example 10: An experiment consists of flipping a fair coin 10 times and counting the number of
heads. Find the mean and standard deviation for this experiment.
success = number of heads.
failure = _______________
p = ½ = 0.5
q = ______
expected # times get heads = mean → μ = n∙p =
standard deviation → σ =
npq =
n = 10
= ____ heads
= _____ heads
Example 11: Forty-three percent of marriages end in divorce. You randomly select 15 married
couples. Find the mean number of marriages that will end in divorce.
n=
success: ______________ , p = _____
failure: _______________
q = ______
μ=
σ=
Class Activity: For the following, first state what n, success, p, failure, and q are. Then answer the
questions
D: Several economics students are unprepared for a multiple-choice quiz with 25 questions, and all of
their answers are guesses. Each question has five possible answers, and only one of them is correct.
a) Find the mean and the standard deviation for the number of correct answers for such students. b)
Would it be unusual for a student to pass by guessing and getting at least 15 correct answers? Why?
or why not?
E: An airline has a policy of booking as many as 21 persons on an airplane that can seat only 20.
Past studies have revealed that only 94% of the booked passengers actually arrive for the flight. Find
the probability that if the airline books 21 persons, not enough seats will be available.
F: A candy company claims that 24% of its plain candies are orange, and a sample of 100 such
candies is randomly selected. Find the mean and standard deviation for the number of orange
candies in such groups of 100.
G: In a study of 420,095 cell phone users in Denmark, it was found that 135 developed cancer of the
brain or nervous system. If we assume that the use of cell phones has no effect on developing such
cancer, then the probability of a person having such a cancer is 0.000340. Assuming that cell phones
have no effect on developing cancer, find the mean and standard deviation for the numbers of people
in groups of 420,095 that can be expected to have cancer of the brain or nervous system.
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TI-84 steps To filling in a binomial distribution function table with n, p known
Go to Table
For L1, Enter in 0, 1, 2, …n
Highlight L2, 2nd, VARS,
Choose Binompdf (n, p). Press Paste/Enter. (2 times)
(Do NOT enter in the third requested value, x)
Your chart should be populated to the number of trials you had.
Finish Kenny’s Notes using this….
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