Chapter 8 Binomial Distributions
Binomial Setting – will have to be able to differentiate between binomial and geometric
#1. Each observation falls into 1 of just 2 categories which for convenience, we will call
“successes” or “failures”
#2. There is a fixed number, n, of observations
#3. The n, observations are all independent  that is, knowing the result of 1
observation tells you nothing about the other observations.
#4. The probability of success, call it p, is the same for each observation
Binomial Random Variable – the # of successes of the random variable X in a binomial
setting
Binomial Distribution - probability distribution of binomial random variable X
** the distribution of the count X, of successes in the binomial setting.
1) has parameters n and p
2) n = # of observations p = the probability of a success on any 1 observation
3) Possible values of X = the whole #’s from 0 to n
4) Notation: X is B(n,p)
***Binomial Random Variables  are discrete !!
Most important skill for using Binomial Distributions  the ability to recognize
situations to which they do and don’t apply.
Binomial Distributions in Statistical Sampling – Steps:
A) Choose an SRS of size n from a population with proportion, p, of successes.
B) When the population is much larger than the sample, the count X of successes in
the sample has approximately the binomial distribution with parameters n and p
(n= the sample size p = to the proportion of successes in the population)
Binomial Formulas – will enable us to find a formula for the probability that a binomial
random variable takes any value by adding probabilities for the
different ways of getting exactly that many successes in n,
observations.
Go over ex. 8.5 p. 517 – need to count the # of arrangements of k success in n
observations
Binomial Coefficient – the # of ways of arranging k successes among n observations is
Without having to list them is Given by:
n
= n!
k
k!(n-k)!
for k = 0,1,2, …n
says: binomial coefficient n choose k
! = factorial - counts the product of the number and numbers below
Ex. 5! = 5∙4∙3∙2∙1 = 120
Binomial Probability – If X has the binomial distribution with n observations and
probability p, of success on each observation, the possible values of X are 0, 1, ..n. If K
is any one of these values:
P(X=k) = ( n )pk(1-p)n-k
(k)
Go over ex. 8.6 p. 519
Finding Binomial Probabilities  rarely will do by hand, a calculator will do it all
Ex. 8.6 p. 520 find probability that no more than 1 of 10 switches in sample fail
inspection B(10,.1)
(p = 1/10 = .10)
P(x≤1) = P(x=1) + P(x=0)
In calc: press 2nd distr binompdf (10,.1,0) + binompdf (10,.1,1)
= .3487 + .3487 = .7361  says that 74% of sample will contain no more than 1
Bad switch
pdf = given a discrete random variable, X, the probability distribution function (pdf)
assigns a probability to each value of X. The probabilities must satisfy the rules for prob.
and given in Ch #6.
Ex. Nightmare from beginning – find the # of correct guesses = 4 (X = 4)
So binompdf(10,.2,4) = .088  8% chance of getting 4 correct
cdf = (cumulative binomial probability) – given random variable, X, the cdf of X
calculates the sum of the probabilities for 0,1,2… up to the value of X.
**That is: it calculates the probability of obtaining at most X successes in n trials
Binomcdf(10,.1,1)  calculates probability x≤1
Can also use Complement Rule – P(x>k) = 1 – P(x≤k)
Binomial Mean and Standard Deviation
1) Mean(μ) = np
n= # of observations
2) Standard Deviation (σ) = √np(1-p)
p = probability of successes
*** Only used for binomial distributions  cannot be used for other discrete random V.
Normal Approximation to Binomial Distributions
** As the # of trials, n, gets larger, the binomial distributions get closer to a Normal Dist.
--- when n is large  we can use Normal probability calculations to
approximate hard to calculate binomial probabilities.
DEF: Normal Approximation for Binomial Distributions
*Suppose that a count X has the binomial distribution with n trials and success
probability, p  When n is large, the distribution of X is approximately normal
N(np, √np(1-p) )

as a rule of thumb – we will use the normal approximation when n and p
satisfy np≥10 and n(1-p)≥10
Go over p. 528 example
Simulating Binomial Events -- in order to simulate must know:
#1. How random variable, X, and success are defined
#2. The probability of success
#3. The number of trials.
Ex. 8.14 simulating on calculator
X = # of hits in 12 free throw attempts is:
P(success) = .75
assign 0 = miss 1 = hit
Randbin(1,.75,12) simulates 12 free throw attempts – selecting 1 75% of the time and
0 25% of the time.
Run – check how many P(x≤7)
Automate more by: 1) assign results to a list and then sum the entries in the list
Randbin(1,.75,12)  L1:sum(L1)
**keep pressing enter until you have 10 #’s – record the #’s
(simulates 10 games)
Geometric Distribution


In a Binomial Distribution  the # of trials is fixed beforehand and the
binomial random variable consists of the # of successes in a fixed # of trials.
In Geometric  are concerned that the random variable X is defined as the #
of counts of the # of trials needed to obtains the 1st success
Geometric Random Variable = defined as the count of the # of trials needed to obtains
the 1st success. It has an infinite set of values b/c it is possible to proceed indefinitely
without ever obtaining a success.
A random variable X is geometric provided the following conditions are met:
Geometric Setting:
#1. Each observation falls into 1 of just 2 categories, which we call “success” or
“failure”
#2. Observations are all independent
#3. The probability of a success, call it p, is the same for each observation
#4. The variable of interest is the # of trials required to obtain the 1st success.
Ex. go over ex. 8.16
Rule for Calculating Geometric Probabilities:
**If X has a geometric distribution with probability p, of success and (1-p) of failure on
each observation, the possible values of X are 1,2,3….If n is any one of these values, the
probability that the first success occurs on the nth trial is: P(X=n) = (1-p)n-1p
** remember the sum of all the probabilities must still = 1
Formula for the sum of Geometric Sequence: Σ p(xi) =
p
1-(1-p)
Do calculator exercise p. 542 8.17
Construct a probability distribution table for x = number of rolls of a die until a 3 occurs:
X
P(X)
1
.1667
2
.1389
3
4
5
6
7
.1157 .0965 .0804 .0670 .0558
Geometpdf(1/6,1)
The Expected Value and Other Properties of the Geometric Random Variable:
Mean of Geometric Random Variable(Expected Value) – if X is a geometric random
variable with probability of success p on each trial, then the mean, or expected value, of
the random variable is: μ = 1/p (the expected # of trials required to get the 1st success)
The variance is: (1-p)
p²
P(X>n) = the probability that it takes more than n trials to see the first success is:
P(X>n) = (1-p)n
Ex. roll a die until a 4 is observed. The probability that it takes more than 6 rolls
To observe a 4 is:
P(X>6) = (1-p)n = (1-1/6)6 = 5/66 = .335
1- geometcdf(1/6,6)
Simulating Geometric Situations – is called “waiting time” simulations b/c you
continue to conduct trials and wait until a success is observed.
Ex. p. 549 Cheerios