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Math 011 – CHAPTER 5 Discrete Probability Distributions
5-2 Probability Distributions
DEFINITION
Random Variable
Probability Distribution
Example: Find the probability for different numbers of girls for couples having two children.
DEFINITION
Discrete Random Variable
Continuous Random Variable
Probability Distribution Requirements
Notation:
Probability Histogram
Example: Does the table of results from a poll describe a probability distribution? Why or why
not?
Should Marijuana use be legal?
Response
P(x)
YES
0.41
NO
DON’T KNOW
0.52
0.07
𝑥
Example: Does P(x) = 3 where x can be 0, 1, 2 determine a probability distribution?
Parameters of a Probability Distribution
MEAN for a probability distribution
VARIANCE for a probability distribution
STANDARD DEVIATION for a probability distribution
Round-off Rule for µ, σ, σ2 from a Probability Distribution
Example: Find the mean, variance, and standard deviation of the probability for different
numbers of girls for couples having two children.
Identifying Unusual Values with the Range Rule of Thumb:
Identifying Unusual Values with Probabilities:
a. x successes among n trials in an unusually high number of successes if the probability of
x or more successes is unlikely with probability of 0.05 or less.
b. x successes among n trials in an unusually low number of successes if the probability of x
or less successes is unlikely with probability of 0.05 or less.
Example: Is 501 heads in 1000 tosses an unusually high number of heads?
Section 5-3
Binomial Probability Distribution
Notation:
Example 1 When an adult is randomly selected (with replacement), there is a 0.85 probability
that this person knows what Twitter is. Suppose that we want to find the probability that
exactly three of the five random adults know what Twitter is.
a. Does the procedure result in a binomial distribution?
b. Id this procedure does result in a binomial distribution, identify the values of n,x,p,q.
Example 2 – In a RingCentral survey, 380 different smartphone users are randomly selected
without replacement. Respondents were asked identify the only thing that they can’t live
without. Responses consist of whether a smartphone was identified. Does this procedure result
in a binomial distribution?
Three Methods for Finding Probabilities
1.
2.
3.
Example 3 (Method 1) Referring to Example 1 part b, use the binomial probability formula to
find the probability of getting exactly three adults who know what Twitter is when five adults
are randomly selected.
Example 4 (Method 2) Assume that when blood donors are randomly selected, 45% of them
have blood that id Group O. If the number of blood donors is n = 20, find the probability that
the number with Group O is x = 16 using StatDisk.
Example 5 (Method 3) Assume that random guesses are made for five multiple choice
questions on an ACT test, so that there are n = 5 trials, each with probability of success (correct)
given by p = 0.20. Use the binomial probability table to find the probability that the number x of
correct answer is fewer than 3.
Example 6 – The display lists the probabilities obtained by entering the values of n = 6 and
p = 0.75 which correspond to the numbers of peas with green pods in a group of six offspring
peas.
a. Find the probability that at most five of the six offspring peas have green pods.
b. Find the probability that at most two of the six offspring peas have green pods. Is two an
unusually low number of peas with green pods? Why or why not?
Peas w/
green pods
x
0
P(x)
0.000
1
0.004
2
0.033
3
0.132
4
0.297
5
0.365
6
0.178
Section 5-4 Parameters for Binomial Distributions
Two Goals of the Section:
For Binomial Distributions (µ,σ2,σ)
Formula 5-6
Formula 5-7
Formula 5-8
Recall: The Range Rule of Thumb
Example 1 The brand name of McDonald’s has a 95% recognition rate. A special focus group
consists of 12 randomly selected adults to be used for extensive market testing.
a. For such random group of 12 people, find the mean and standard deviation for the
number of people who recognize the brand name of McDonald’s.
b. Use the range rule of thumb to find the minimum usual number and the maximum usual
number of people who recognize the brand name of McDonald’s.
c. In one particular randomly selected group of 12 people, is 9 an unusually low number of
people who recognize the brand name of McDonald’s? Is 12 an unusually high number
of people?
Example 2 Among 945 births, there were 879 girls born to parents using the XSORT method of
gender selection.
a. Assuming that boys and girls are equally likely, use Formula 5-6 and 5-8 to find the mean
and standard deviation for the numbers of girls born in groups of 945 babies.
b. Use the range rule of thumb to find the minimum and maximum usual number of girls
born in groups of 945 births, assuming that boys and girls are equally likely.
c. The Chapter Problem described the XSORT method of gender selection that resulted in
879 girls born in a group of 945 babies. Is 879 girls unusually high? Interpret the findings
Example 3 Mars, Inc. claims that 20% of its M&M plain candies are orange, and a sample of 100
such candies is randomly selected.
a. Find the mean and standard deviation for the number of orange candies in such groups
of 100.
b. Data set 20 in Appendix B consists of a random sample of 100 M&Ms, including 25 that
are orange. Is this result unusually high? Does it seem that the claimed rate of 20% is
wrong?