Download Use Normal model

Chapter 17 Part 2
Binomial Model
Combinations
The number of ways we can have k successes in
n trials is called a combination.
𝑛
𝑛!
𝑜𝑟 𝑛 𝐶𝑘 =
𝑘
𝑘! 𝑛 − 𝑘 !
Combinations
Example: How many ways can I choose two
students from a class of 30 to be class
representatives?
30
𝑜𝑟
2
30!
= 435
30 𝐶2 =
2! 30 − 2 !
Combinations
On the calculator:
Type 30, MATH, PRB, nCr, 2, ENTER
30 → 𝑀𝐴𝑇𝐻 → 𝑃𝑅𝐵 → 𝑛𝐶𝑟 → 2
= 435
Binomial Model
If we want to know the probability of exactly k successes in n
Bernoulli trials, we use the Binomial Model:
p = probability of success
q = probability of failure
𝑛 𝑘 𝑛−𝑘
𝑝 𝑞
𝑘
Binomial Model
𝑚𝑒𝑎𝑛 𝜇 = 𝑛𝑝
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝜎 =
𝑛𝑝𝑞
Example:
Suppose 20 donors come to a blood drive and that 6% of
people are universal donors (type O-negative). What is the
probability that there are 2 universal donors present?
20
2
18
𝑃 𝑋=2 =
(.06) (.94) = 0.2246
2
Example:
Suppose 20 donors come to a blood drive and that 6%
of people are universal donors (type O-negative).
What is the mean and standard deviation of number
of universal donors?
𝜇 = 𝑛𝑝 = 20 .06 = 1.2
𝜎 = 𝑛𝑝𝑞 =
20 .06 (.94) = 1.06
Normal Model
When the numbers are just too big, we can use
the Normal model.
Example: What is the probability of getting at
least 1850 units of O-negative blood from 32,000
donors?
It is extremely difficult to find P(X=1850) +
P(X=1851) + P(X=1852) + … to infinity.
Normal Model
Example: What is the probability of getting at
least 1850 units of O-negative blood from 32,000
donors?
Recall that p = 0.06
𝜇 = 32000 .06 = 1920
𝜎=
32000 .06 (.94) = 42.28
Normal Model
Example: What is the probability of getting at
least 1850 units of O-negative blood from 32,000
donors?
Now find the z-score
1850 − 1920
𝑧=
= −1.65
42.28
Normal Model
Example: What is the probability of getting at
least 1850 units of O-negative blood from 32,000
donors?
Now use z-table to find P(z > -1.65)
𝑃 𝑧 > −1.65 = 1 − 0.05 = 0.95
Normal Model
Success/Failure Condition
In order to use the Normal model for a
binomial distribution, we must check that
two things are true:
np≥ 10 and nq≥ 10
Normal Model
Success/Failure Condition
From the last example, p=0.06, q=0.94,
and n=32,000
32000(.06) ≥ 10 and 32000(.94) ≥ 10
Tip:
Use the Normal model to answer questions
asking for “at least” if numbers are large.
Use the complement rule to answer
questions asking for “at least” if numbers
are small.
TI Shortcut:
2nd → 𝐷𝐼𝑆𝑇𝑅 → 𝑏𝑖𝑛𝑜𝑚𝑝𝑑𝑓 𝑛, 𝑝, 𝑋
2𝑛𝑑 → 𝐷𝐼𝑆𝑇𝑅 → 𝑔𝑒𝑜𝑚𝑒𝑡𝑝𝑑𝑓(𝑝, 𝑋)
Example:
What is the probability of finding at least
7360 cars in the state that are a Saturn?
Use Normal model
What is the probability of getting at least 3
blue M&M’s in a bag?
Use 1-P(X=1 or 2)
Assume that 13% of people are left-handed. If
we select 5 people at random, find the
probability that:
a) There are at least 3 lefties in the group
b) There are no more than 3 lefties in the group
Assume that 13% of people are left-handed. If
we select 5 people at random, find the
probability that:
a) P(X=3)+P(X=4)+P(X=5) =0.0179
or 1-[P(X=1)+P(X=2)]
b) P(X=0)+P(X=1)+P(X=2)+P(X=3) =0.9987
or 1-[P(X=4)+P(X=5)]
Assume that 13% of people are left-handed.
Suppose we choose 12 at random.
a)Find the mean and standard deviation of the
number of right-handers in the group.
b)What’s the probability that there are exactly 6
lefties and 6 righties?
c)What’s the probability that the majority is
right-handed?
Assume that 13% of people are left-handed.
Suppose we choose 12 at random.
a) np=12(0.87) = 10.44; SD= 12 .87 (.13) = 1.16
b)P(X=6)=0.00193
c)P(X>6)=P(7)+P(8)+…+P(12) = 0.998
Today’s Assignment:
 All Chapter 17 Homework due
Friday
 Unit 4 test Tuesday