Download Probability Distributions - Valdosta State University

Probability Distributions and
the Binomial Distribution
BUSA 2100, Sections 5.1 - 5.4
Random Variables
A random variable is a quantity that
changes values from one occurrence to
another, in no particular pattern.
 Example 1: Let X = number of heads
on 3 tosses of a coin; X = 0, 1, 2, 3 .
 Example 2: Let Y = number of papers
sold daily in a newspaper rack;
Y = 0, 1, 2, 3, 4, 5 .

Discrete and Continuous
Random Variables
A discrete random variable can have
only a small number of specific possible
values (usually whole numbers). It tells
“how many.” The random variables in
Examples 1, 2 are discrete.
 A continuous random variable can
have an undetermined large finite
number of values. It tells “how much.”

Continuous Random Variables
Example 3: Let W = amount of time in
minutes that a family watches TV in a
month. This is a continuous random
variable.
 Example 4: Weight and distance are
also continuous random variables.

Probability Distributions
A
probability distribution (for a
discrete random variable) is a list of
all possible outcomes together with
their associated probabilities.
 State probability distribution for 3
coins.
Expected Value
What is the “average” value for X for the
probability
X
P(X)
distribution
60 .1
shown at
70 .2
the right?
80 .3
90 .4
 The median is 75; the simple average is
75; is there a more accurate answer?

Expected Value, Page 2

.
Republican/Democrat Ex.

Ex. 1A: A large group of people consists
of 40% Republicans and 60% Democrats. If a sample of 5 is chosen, what is
the probability of getting 2 Republicans
and 3 Democrats, in that order?
Rep/Dem Example, Page 2

Ex. 1B: What is the prob. of choosing 2
Repub. and 3 Democrats in any order?
Rep/Dem Example, Page 3

To avoid listing the number of ways that 2 R’s
and 3 D’s can be arranged, we can use
combinations to determine how many ways
we can choose which two of the 5 selections
will be Republicans.

The Republican/Democrat problem is an
example of the binomial distribution.
Intro. to Binomial Distribution
Binomial problems have 2 characteristics
(requirements).
 (1) Most important characteristic: Each
selection must have exactly two possible
outcomes.
 Examples:


(2) Each selection is independent of the
other selections.
Binomial Notation & Formula

Do each item for Rep./Dem. problem.
n = number of selections
Define “success”.
p = probability of success on one selection

r = number of successes



Binomial Problems, Page 1


Example 1: A large lot of manufactured
items contains 10% defectives. In a
random sample of 6 items, what is the
probability that exactly 2 are defective?
Binomial Problems, Page 2

Example 2: Thirty percent of customers
that enter an appliance store make purchases. What is the prob. that 4 of the
next 10 customers will buy something?
Binomial Problems, Page 3
For convenience we will use a binomial
table, looking up n, p, r, in that order.
 Example 3A: A large lot of manufactured items contains 20% defectives. In
a random sample of 8 items, what is the
probability that 5 or more items are
defective?

Binomial Problems, Page 4

Ex. 3B: P(2 or fewer) =
Binomial Problems, Page 5

Example 4: Sixty percent of the workers
in a plant belong to a union. A random
sample of 12 is chosen. Find the probability that exactly 4 belong to a union.
Binomial Problems, Page 6

.
Binomial Problems, Page 7

Example 5: At Blaylock Company, in the
past, 25% of new employees were not
hired for a permanent position after a
six-months probationary period. Among
7 new employees, what is the prob. that
5 or more will be hired permanently?
Binomial Problems, Page 8

It is essential that the values of p and r
are consistent with the way that success
is defined.