Computing Binomial ProbabilitiesA Stats 10 test has 4 multiple choice questions with four choices with one correct answer each. If we just randomly guess on each of the 4 questions, what is the probability that you get exactly 1 question correct?
a) 0.04668 b) 0.42188 c) 0.10547 d) 0.25

Binomial Distribution Function
•
The formula that finds the probabilities for the binomial
distribution for probability of success p, fixed number of
trials n, and k successes is as follows:

Binomial Coefficient
•
The n over the k inside the parentheses can be read as “n
choose k”
•
Instead of writing all different combinations of outcomes
and counting them all one-by-one this provides us the
number of all those combinations.

Factorials
•
! - indicates a
factorial
•
n! = n x (n-1) x (n-2) x (n-3) x
....
x 1
5! = 5 x 4 x 3 x 2 x 1 = 120
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

Binomial Coefficient Examples

Binomial Coefficient Hints

Computing Binomial Probabilities
•
A Stats 10 test has 4 multiple choice questions with four
choices with one correct answer each. If we just randomly
guess on each of the 4 questions, what is the probability
that you get exactly 2 questions correct?
•
Using the binomial probability function:

Computing Binomial Probabilities•A Stats 10 test has 5 multiple choice questions with four choices with one correct answer each. If we just randomly guess on each of the 5 questions, what is the probability that you get exactly 2 questions correct?

Computing Binomial Probabilities
•
A Stats 10 test has 5 multiple choice questions with four
choices with one correct answer each. If we just randomly
guess on each of the 5 questions, what is the probability
that you get 4 or more questions correct?

Computing Binomial Probabilities•A Stats 10 test has 5 multiple choice questions with four choices with one correct answer each. If we just randomly guess on each of the 5 questions, what is the probability that you get at least 1 question correct?

Expected Value and Standard Deviation
•
The mean and standard deviation of the binomial can be
easily calculated
•
Their interpretation is the same as with all distributions.
Mean is the center and standard deviation tells us how far
values typically are from the mean.
•
Expected value or Mean = np
•
Standard deviation =

Expected Value Example
•
A Stats 10 test has 4 multiple choice questions with one
correct answer each. If we just randomly guess on each of
the 4 questions, what is the expected number of questions
we get correct?
•
Expected value = np = 4 x 0.25 = 1
•
Standard deviation =
= 0.866
•
We are expected to only get 1 out of 4 questions correct
if we just randomly guess.

Continuous Probability Distribution
Function
•
Often represented as a curve
•
The area under the curve between two values of x
represents the probability of x being between the two
values
•
The total area under the curve must equal 1
•
The curve cannot lie below the x-axis

•
The
Normal Model
is a good fit if:
•
The distribution is unimodal
•
The distribution is approximately symmetric
•
The distribution is approximately bell shaped
•
A Normal distribution is defined by the mean
and
standard deviation
. Shorthand for a normal distribution is
N(
,
)
•
The Normal distribution is also called the Gaussian
distribution or the Bell Curve
The Normal Model

Standardizing with z-scores
•
Reminder: z-scores are standardized scores
•