Warm-up

Grab a die and roll it 10 times and record
how many times you roll a 5. Repeat this
7 times and record results.

This time roll the die until you get a 5.
Record how many rolls it took you.
Repeat this 7 times and record results.
Warm-Up
What’s the smallest number you can get
for the first one?
 What’s the largest number?

What’s the smallest number you can get
for the second one?
 What’s the largest number?

6.2 Homework Questions
Section 6.3
Binomial Random Variables
What does “bi” mean?

List all of the words you can think of that
start with “bi”…
Binomial Setting

The four conditions for a binomial setting are:
Success/Failure
2. Independent Trials
3. Constant “p” (probability of success)
4. Set number of trials, n
1.
Geometric

1.
2.
3.
4.
The four conditions for a geometric
setting are:
Success/Failure
Independent Trials
Constant “p” (probability of success)
No set number of trials, n
Binomial Random Variable

The count X of successes in a binomial
setting is a binomial random variable. The
probability distribution of X is a binomial
distribution with parameters n and p. The
possible values of X are the whole
numbers from 0 to n.
Binomial?
Genetics says that children receive genes from
each of their parents independently. Each child of
a particular pair of parents has probability 0.25 of
having type O blood. Suppose these parents have
5 children. Let X = the number of children with
type O blood.
 Shuffle a deck of cards. Turn over the first 10
cards, one at a time. Let Y = the number of aces
you observe.
 Shuffle a deck of cards. Turn over the top card.
Put the card back in the deck, and shuffle again.
Repeat this process until you get an ace. Let W =
the number of cards required.

Binomial Probabilities
Let’s do the children’s gene problem…
 P(none of the children have type O)=


P(x=1)
Building the formula
P(x = k) = P(exactly k successes in a trial)
= number of arrangements∙ 𝑝𝑘 (1 − 𝑝)𝑛−𝑘

Number of arrangements:
Binomial Coefficient




The number of ways of arranging k successes
among n observations is given by the
binomial coefficient:
𝑛
𝑛!
=
𝑘! 𝑛−𝑘 !
𝑘
Do you remember what n! means?
𝑛
𝑛
CAUTION has nothing to do with
𝑘
𝑘
For example…

When x = 1, we had 5 arrangements…
There is a button on your calculator! 
 5 nCr 1
 Math – Prob – nCr

Binomial Probability

If X has the binomial distribution with n
trials and probability p of success on each
trial, the possible values of X are
0, 1, 2,…, n. If k is any one of these values,
𝑛 𝑘
P X=k =
𝑝 (1 − 𝑝)𝑛−𝑘
𝑘
 This is on the formula sheet! 
Example

Find the probability that exactly 3
children have type O blood.

Should the parents be surprised if more
than 3 of their children have type O
blood? Justify your answer.
Mean and Standard Deviation of a
Binomial Distribution

Blood Type Probability Distribution:
X
0
1
2
3
4
5
P(X)
0.23730
0.39551
0.26367
0.08789
0.01465
0.00098
Mean and Standard Deviation of
Binomial Random Variables
If a count X has the binomial distribution
with number of trials n and probability of
success p, the mean and standard
deviation of X are:
 𝜇𝑥 = 𝑛𝑝

 𝜎𝑥
= 𝑛𝑝(1 − 𝑝)
 Remember – these formulas ONLY work
for binomial distributions!
Homework #3
Together, let’s do numbers 69-72
 Pg. 403 (73-75, 77, 79, 80, 82, 84-87, 89-92, 94-105)

Warm-Up
Normal Approximation for Binomial
Distributions


As a rule of thumb, we will use the
Normal approximation when n is so large
that:
𝑛≤
1
𝑁
10
𝑛𝑝 ≥ 10
 𝑛(1 − 𝑝) ≥ 10
 That is, the expected number of successes
and failures are both at least 10.

Example
Suppose that exactly 60% of all adult US
residents would say “agree” if asked if they
think shopping is frustrating. A survey asked
nationwide sampled 2500 adults.
 Let X = the number of people who agree.

◦ Show that X is approximately a binomial
random variable.
Example

Check the conditions for using a Normal
approximation in this setting.
Example

Use a Normal distribution to estimate the
probability that 1520 or more of the
sample agree.
Homework #3
Together, let’s do numbers 69-72
 Pg. 403 (73-75, 77, 79, 80, 82, 84-87, 89-92, 94-105)
