5-2 Probability Models
The Binomial Distribution and
Probability Model
Binomial Experiments
Also called Bernoulli Experiments
Imagine those boxes of Wheaties again
that have pictures of sports stars in
them. What if you only want a picture
of Tiger Woods? That is, opening a box
with Lance or Serena is no good, but
Tiger is good?
Binomial Experiments (cont)
A Binomial experiment has a fixed number of
trials (n) which are independent and
repeated under identical conditions.
Each trial has only one of two outcomes –
Success or Failure
The probability of success (p) or failure (q) is
the same for every trial
The central question will always be: What is
the probability of r number of successes out
of n trials
Binomial Experiments (cont)
Other Binomial (Bernoulli) Trials are
- Tossing a coin
- Drawing a card (with replacement)
- Finding a defective product on an
assembly line
- Tossing a ring onto bottles at Point
Pleasant
Binomial Experiments (cont)
How do we find p? Hmmmm…
One note: If the selection is made without replacement, is that
independent?
If the number of trials is small enough with respect to the
population, it is considered to be essentially independent, and
a Binomial Experiment can be used to approximate the
outcome.
The value of p will round to be close enough. The book has a
good discussion of this.
How does this work?
Imagine tossing a coin three times and getting heads
exactly two times.
How could this happen?
Imagine all outcomes
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
No heads: 1
One heads: 3
Two heads: 3
Three heads: 1
What if we toss more times?
Toss that coin four times
HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT
THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT
No heads: 1
One heads: 4
Two heads: 6
Three heads: 4
Four heads: 1
Do those numbers look familiar?
Binomial Experiments (cont)
This will look familiar…
A binomial experiment is fashioned on a
binomial expansion that looks like this:
Binomial Experiments (cont)
This will look familiar…
A binomial experiment is fashioned on a
binomial expansion that looks like this:
n
(failure  success)
Binomial Experiments (cont)
From last year you learned how to
expand using the shortcuts.
The application here is that the formula
for a binomial probability from this
formula is given by
Binomial Experiments (cont)
From last year you learned how to
expand using the shortcuts.
The application here is that the formula
for a binomial probability from this
formula is given by
r
n r
P(r)  n Cr p q
p = probability of success; q = failure; r = number of successes in n trials
A few notes…
1. If you are looking for x or fewer (or x
or more) successes, you need to
compute P(x) + P(x-1) + P(x-2)…
That is the probability of tossing 3 or
fewer heads = P(3) + P(2) + P(1) + P(0)
A few notes…
2. Guess what – the calculator can
compute binomial probabilities!
Hallelujah!!
MATH: PRB: 3:nCr
Put the n in before you press this button,
then press r. Lets try to find 12C2
DISTR: 0: binompdf(n,p,r) gives P(r)
DISTR: A: binomcdf(n,p,r) gives P(x ≤ r)
A few notes
There is also a table in the back that you
should try with one (any one) problem.
This table, table 3 in Appendix II has n
from 2 to 20 computed probability
values for certain p values.
A few notes…
3. μ = np and σ  npq for a binomial
probability model. Remember that μ is
also called the expected value
A few notes…
4. The model to decide the number of
trials till success is called a Geometric
Probability Model for Bernoulli, and the
expected value here is
A few notes…
4. The model to decide the number of
trials till success is called a Geometric
Probability Model for Bernoulli, and the
expected value here is
1
μ
p
1 p
q
and σ 

2
p
p
A few notes…
4. The model to decide the number of
trials till success is called a Geometric
Probability Model for Bernoulli, and the
expected value here is
1
μ
p
1 p
q
and σ 

2
p
p
The chance of success on the xth trial is
A few notes…
4. The model to decide the number of
trials till success is called a Geometric
Probability Model for Bernoulli, and the
expected value here is
1
μ
p
1 p
q
and σ 

2
p
p
The chance of success on the xth trial is
P(X  x)  qx 1p
Think about last chapter and basic
probability… this is based on order
mattering
A few notes…
4. Success/Failure condition – A
binomial model is described as
approximately Normal if np ≥ 10 and
np ≥ 10.
We’ll discuss that more in the future as
well.
Lets try a problem
Suppose 20 donors come in for a blood drive.
A) If donors line up at random, how many do you
expect to examine before you find someone who is
a universal donor? Universal donors are O negative.
Only about 6%
B) what is the probability that the first Universal donor
is one of the first four people in line? *
* This means
probability that
thethere
first person
C) What
is thethe
probability
that
are is2plus
or 3the
probability
that the first person is not but the second is, plus the
universal
donors?
probability that the first and second are not but the third is… D) What
are
the mean and standard deviation of the
like last
chapter
number of universal donors among them?