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Math 123- Statistics
Chapter 4 Notes
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4.1 Probability Distributions
Recall: For each probability problem, first write the probability notation, then do the problem.
Def- A random variable, x, represents a numerical value associated with each outcome of a
probability experiment.
Def- A discrete random variable has a countable number of possible outcome that can be listed.
Def- A continuous random variable has an uncountable number of outcomes. (Sometimes
represented by an interval on a number line.)
Ex- Determine whether the random variable is discrete or continuous.
a) x is the number of pumps in use at a gas station.
b) x represents the weight of a truck at a weigh station.
Def- A discrete probability distribution lists each possible value that the random variable can
assume, together with its probability. Each probability is between 0 and 1, inclusive, and the sum of
the probabilities is one.
Ex- Construct a discrete probability distribution for the following. The data includes the number of
packs of gum purchased.
# packs of gum 0
1
2 3 4
Frequency
17 11 9 5 17
Ex- Determine if the distribution is a discrete probability distribution. If not, identify the property that
is violated. The daily limit for catching bass at Lopez Lake is 4. x=number of fish caught in a day
x
0
1
2
3
4
P(x) .36 .23 .08 .14 .29
Def- The mean of a discrete random variable is    xP(x) . Note: This mean represents the
“theoretical average” after performing the experiment many thousands of times.
Def- The variance of a discrete random variable is  2   ( x   ) 2 P( x) .
Def- The standard deviation of a discrete random variable is  
 (x  )
2
P( x) .
Ex- Construct a discrete probability distribution, then find the mean, variance, and standard
deviation of the discrete probability distribution.
The number of goals scored by a soccer team during a 30-game season is given below.
# goals
0
1
2
3
4
5
# games 1
9
11
4
3
2
Def- The expected value of a discrete random variable is equal to the mean of the discrete random
variable.
  E( x)   xP( x)
Ex- The probability distribution shows the number of pints of blood that people donating blood give
each year along with the probability. Find the expected value and the standard deviation of the
discrete probability distribution. Explain what each of these mean in the context of the problem.
# pints/yr 1
2
3
4
5
6
P(x)
.71 .12 .03 .1 .03 .01
Ex- You have the opportunity to receive a gift. Use the idea of an expected value to determine
which gift would be worth more money.
Choice A: The gift is $240.
Choice B: The gift is a 25% chance to win $1000.
Ex- You are interested in purchasing car insurance. Use the idea of an expected value to determine
if purchasing car insurance is really worth the money.
Choice A: Purchase car insurance for $740. (This is a sure/certain loss of $740.)
Choice B: Do not purchase car insurance. (It costs $7,500 if you get in an accident and there is a
10% chance that you will get in a car accident.)
4.2 Binomial Distributions
Def- A binomial experiment is a probability experiment that satisfies the following conditions:
1. The experiment is repeated for a fixed number of trials where each trial is independent of
the other trials.
2. There are only two possible outcomes of interest, success S and failure F.
3. The probability of success, P(S) is the same for all trials.
4. The random variable x is the number of successful trials.
Notation:
n= # trials
p= P(S) = probability of success
q= P(F) = probability of failure
q= 1 – p
x= # successes in n trials
x= 0, 1, 2, 3, …, n
Ex- The following situation is binomial. A person chooses one pencil at a time from a large box,
records if the pencil is blue or not, then replaces the pencil back in the box.
Ex- Match the probabilities p= .85, p= .35, p= .5 with the graphs.
Ex- Match the graphs with the given values of n; n= 5, n= 18, n= 7.
Binomial Probability Formula- The probability of exactly x successes in n trials is P( x) n C x p x q n  x .
Ex- For past records, a clothing store finds that 30% of people who enter the store make a
purchase. The store manager is interested in the 15 people currently in the store.
a) State n, p, q and possible values of x for this situation.
b) Find the probability that exactly 4 people do not buy anything in the store.
c) Find the probability that exactly 4 people buy something in the store.
Ex- Forty-three percent of U.S. adults receive calls from telemarketers. In a random sample of 7
adults, what is the probability that the number of people receiving calls from telemarketers is …
a) exactly 3.
b) between 2 and 5.
c) more than 3.
Finding a Binomial Probability Using the Calculator
Single-term binomial P(x=r)
Cumulative binomial P ( x  r )
nd
2 Distr binompdf(n, P(x), r)
2nd Distr binomcdf(n, P(x), r)
Finding a Binomial Probability Using a Table p. A8 – A10
x
0
1
2
3
4
p
.6
.001
.008
.041
.124
.232
5
6
7
8
.279
.209
.090
.017
For n=8 and P(x)=.6
P(x=5)=
P(x  5) 
Population Parameters for a Binomial Distribution
Mean   np
Variance  2  npq
Std. Dev.   npq
Ex- Find the mean, variance, and std. dev. for a binomial data set of 13 data values where the
probability of success is .74.
Mean
Variance
Std. dev.
4.3 More Discrete Probability Distributions
Def- A geometric distribution is a discrete probability distribution of a random variable x that satisfies
the following conditions:
1. The trial is repeated until a success occurs.
2. The repeated trials are independent of each other.
3. The probability of success, p, is constant for each trial.
Geometric Probability Formula- The probability that the first success will occur on trial number x is
P( x)  pq x 1 where q= 1 – p.
Note: x is defined when x= 1, 2, 3, … x  0 because you cannot have a success if you haven’t had
a trial yet.
Ex- Find the indicated probabilities using the geometric distribution. During a promotional contest, a
soft drink company places winning caps on one out of every six bottles. Find the probability that you
find your first winning cap…
a) before the fourth soft drink bottle you purchase.
b) within the first four soft drink bottles you purchase.
c) after the third soft drink bottle purchased.
Def- The Poisson distribution is a discrete probability distribution of a random variable x that
satisfies the following conditions:
1. The experiment consists of counting the number of times, x, an event occurs in a given
interval. The interval can be an interval of time, area, volume, etc.
2. The probability of the event occurring is the same for each interval.
3. The number of occurrences in one interval is independent of the number of occurrences in
other intervals.
 x e 
Poisson Probability Formula- The probability of exactly x occurrences in an interval is P( x) 
x!
where  is the mean number of occurrences per interval unit. Note: x= 0, 1, 2, ….
Ex- Find the indicated probabilities using the Poisson distribution. During a 36-year period, lightning
killed an average of 3239 people in the U.S. Assume that this rate continues to hold true
indefinitely. Find the probability that in a 36-year period…
a) 2800 people in the U.S. will be struck and killed by lightning.
b) between 3000 and 3300 people in the U.S. will be struck and killed by lightning.
c) at least 3300 people in the U.S. will be struck and killed by lightning.
Finding a Poisson Probability Using a Table p. A11 – A15
For  =8.3 and x=5 portion of table shown.

x
0
1
2
3
4
5
6
7
8.3
.0002
.0021
.0086
.0237
.0491
.0816
.1128
.1338
P(x=5)=
P(x  5) 
Note: There aren’t geometric tables because all functions are just on the calculator and number of
trials for first success are never-ending. (So the table would have to be infinite.)
Finding Probabilities Using the Calculator
Single-term Binomial P(x=r)
2nd Distr binompdf(n, P(x), r)
Cumulative Binomial P ( x  r )
2nd Distr binomcdf(n, P(x), r)
Single-term Geometric P(x=r)
2nd Distr geometpdf(P(x), r)
Cumulative Geometric P ( x  r )
2nd Distr geometcdf(P(x), r)
Single-term Poisson P(x=r)
2nd Distr poissonpdf(  , r)
Cumulative Poisson P ( x  r )
2nd Distr poissoncdf(  , r)
Calculator Practice
Binomial distribution where n=18 and p= .51.
P (x  13) 
P(5  x  12) 
Geometric distribution where p= .62.
P(x  4) 
P(x  7) 
Poisson distribution where   7.3 .
P(x  5) 
P(x  9) 