Chapter 17
Special Discrete
Distributions
Binomial Distribution B(n,p)
• Each trial results in one of two
mutually exclusive outcomes.
(success/failure)
• There are a fixed number of trials
• Outcomes of different trials are
independent
• The probability that a trial results in
success is the same for all trials
• The binomial random variable x is
defined as the number of successes out
of the fixed number
Are these binomial distributions?
1) Toss a coin 10 times and count the
number of heads
Yes
2) Deal 10 cards from a shuffled deck
and count the number of red cards
No, probability does not remain constant
3) Two parents with genes for O and A
blood types and count the number of
children with blood type O
No, no fixed number
Toss a coin 3 times and count the
number of heads
Find the discrete probability distribution
X
P(x)
0
1
2
3
.125
.375
.375
.125
Out of 3 coins that are tossed, what is
the probability of getting exactly 2 heads?
Binomial Formula:
n  k
n k
P (x  k )    p 1  p 
k
 
Where:
n 
 n C k
k 
Out of 3 coins that are tossed,
what is the probability of
getting exactly 2 heads?
3 2
1
P (x  2)   0.5 0.5  .375
2
The number of inaccurate gauges in a
group of four is a binomial random
variable. If the probability of a defect
is 0.1, what is the probability that only
1 is defective?
 4 1
3


P (x  1)   0.1 0.9  .2916
1 
More than 1 is defective?
P (x  1)  1  (P (0)  P (1))  .0523
Calculator
Steps
The calculator will calculate the probability of
a number of success in n trials.
• 2nd DISTR – binompdf(n,p,x)
The calculator will also calculate the
cumulative probability.
• 2nd DISTR – binomcdf(n,p,x)
– This calculates the probability of getting x or fewer
successes among the n trials.
Binomial formulas for mean
and standard deviation
 x  np
 x  np 1  p 
In a certain county, 30% of the
voters are Republicans. How
many Republicans would you expect
in ten randomly selected voters?
What is the standard deviation for
this distribution?
 x  10(.3)  3 Republicans
x  10(.3)(.7)  1.45 Republicans
Geometric Distributions:
• There are two mutually exclusive
So what are the
outcomesHow far
this go?
To will
infinity
possible values of X
• Each trial is independent of the
others
• The probability of success
remains constant for each trial.
1 2 variable
3 4 .x .is.the
•X
The random
number of trials UNTIL the
FIRST success occurs.
Differences between binomial
& geometric distributions
• The difference between
binomial and geometric
properties is that there is
NOT a fixed number of
trials in geometric
distributions!
Geometric Formulas:
P ( x)  1  p 
1
x 
p
1 p
x 
2
p
x 1
p
Not on formula
sheet – they will
be given on quiz or
test
A real estate agent shows a house to
prospective buyers. The probability that
the house will be sold to the person is
35%. What is the probability that the
agent will sell the house to the third
person she shows it to?
P  X  x   1  p 
x 1
p
.65
31
.35
.1479
How many prospective buyers does she
expect to show the house to before
someone buys the house? SD?
1
1  .35
x 
 2.86 buyers  x 
 2.304 buyers
2
.35
.35