Download Notes Chapter 14

Notes Chapter 14
Probability is the branch of mathematics that describes the pattern of chance outcomes.
Random is a description of a kind of order that emerges only in the long run. Individual
outcomes are uncertain but there is a regular distribution of outcomes in a large number of
repetitions.
The probability of any outcome is the proportion of times the outcome would occur in a very
long series of repetitions.
For any random phenomenon, each attempt, or trial, generates an outcome.
The sample space, S, of a random phenomenon is the set of all possible outcomes for a variable.
(Be sure you know what variable you are representing.)
Sampling can be done with or without replacement. Determining which to use is generally
done from the context of the problem.
An event is an outcome or a set of outcomes of a random phenomenon, a subset of the sample
space.
Disjoint events, also called mutually exclusive, have no outcomes in common. They cannot
occur at the same time.
Events are independent (informally) when the outcome of one event does not influence the
outcome of any other event.
Be certain not to confuse disjoint with independent. Disjoint events cannot occur together;
therefore they cannot be independent.
The Law of Large Numbers says that the long-run relative frequency of repeated independent
events settles down to the true probability as the number of trials increases.
A probability distribution for a certain variable is made up of the sample space, S, and the
individual probabilities of each event in the sample space.
The following three rules apply to probability distributions. S represents the sample space and x
any event in the sample space.
Rule 1: The probability of any event is a number between 0 and 1.
0  P(x)  1.
Rule 2: The sum of all possible outcomes must equal 1
P(S) = 1.
Rule 3: The probability that an event does not occur is 1 minus the probability
that the event does occur. This is called the probability of the complement, Ac.
P(Ac) = 1 – P(A)
Notes Chapter 15
The union (U) of any collection of events is the event that at least one of the collection occurs.
Union of two events is when either one or the other, or both events occur.
The intersection (∩) of any collect of events is when all the events occur. Intersection of two
events is when both event A and event B occur.
Conditional Probability:
P(B|A)
This is read as “the probability of B given A.
It is the conditional probability that event B occurs given that event A occurs.
There are two rules of probability that are given on the AP Statistics formula sheet.
1. General Addition Rule for Union of Two Events
For any two events A and B,
P(A U B) = P(A) + P(B) – P(A ∩ B)
The second stems from the general multiplication rule which states:
General Multiplication Rule
P(A ∩ B) = P(B)P(A|B)
Based on the last formula we can define a rule for conditional probability:
2. Definition of Conditional Probability
When P(A) > 0,
P(A|B) = P(A ∩ B)
P(B)
Independent Events – formally
Two events A and B that both have positive probability are independent if
P(B|A) = P(B)
In other words, the probability of event B is not affected by the event A. This fits with the
informal definition of independence but now gives us a way to show two events are independent
mathematically. This formula is not on the AP Stat formula chart.
Notes for Chapter 16
Discrete and Random Variables
A random variable is a variable whose value is a numerical outcome of a random phenomenon.
In this section we will see two ways of assigning probabilities to the values of a random variable.
Discrete Random Variables
A discrete random variable X has a countable number of possible values. The probability
distribution of X lists the values and their probabilities.
Value of X x1 x2 x3 … xk
Probability p1 p2 p3 … pk
The probability pi must satisfy the rules of probability.
We can use a probability histogram to picture the probability distribution of a discrete
random variable.
The mean of a random variable X (x)is like a weighted average that takes into account that all
the outcomes are not equally likely. x is called the expected value. That does not mean we
expect to get that value on any one trial, it means that we expect to get that average value,
overall, in the long run.
Given a Discrete Probability Distribution:
Value of X x1 x2 x3 … xk
Probability p1 p2 p3 … pk
We do a weighted average to find the mean
x = x1p1 + x2p2 + … + xkpk = xipi
The variance of X is
2
2
2
X = (x1 - x) p1 + (x2 - x) p2 + … +
(xk - x)2pk = (xi - i)2pi

The Standard Deviation (x) of X is the square root of the variance.
Both of the formulas above are on the AP Stat Formula Chart.
Rules for Means of Distributions:
1) If X is a random variable and you wish to transform the data with an equation, you may apply
the same equation you would use on the data to the old mean to create a new mean.
2) If X and Y are random variables and you wish to combine or take the difference of their
distributions to create a new distribution, then you either combine or take the difference
(respectively) of their means for a new distribution.
Rules for Variances of Distributions:
1) If X is a random variable and you wish to transform the data with an equation, a) disregard
any constant values in the equation as they will not affect the variance b) to get the variance of
the new distribution, multiply the old variance by any coefficient squared.
2) If X and Y are independent random
variables and you wish to create a new distribution by either adding or subtracting the old
distributions, then you may add (for sums & differences) the old variances.
Remember: Variances of Random variables, add; standard deviations do not.
Examples:
Brown Hair/ Brown Eyes
In a certain town, 40% of the people have brown hair, 25% have brown eyes, and 15% have both
brown hair and brown eyes. A person is selected at random from the town.
1. If he or she has brown hair, what is the probability that he or she also has brown eyes?
2. If he or she has brown eyes, what is the probability that he of she does not have brown hair?
3. What is the probability that he or she has neither brown hair or brown eyes?
Birthday Gifts
In a book titles, Birthday Gifts, Brier and Rogers found that 71% of married men will get flowers
or buy jewelry for their wives on their birthdays. 58% of married men will buy jewelry and 39%
will buy flowers for their wives. What is the probability that a married man will buy flowers and
jewelry for his wife’s birthday?
Oh, Say can you see?
In the book Chances: Risk and Odds in Everyday Life, James Burke says that 56% of the general
population wears eyeglasses, while only 3.6% wears contacts. He also says that of those who
wear glasses, 55.4% are women. Of those who wear contacts, 36.9% are men. Assume that no
one wears both glasses and contacts. For the next person you encounter at random, what is the
probability that this person is:
A) not wearing contacts
B) a woman wearing glasses
C) a man wearing contacts
Blood Example (Book BVD p. 341 #24)
The American Red Cross says that about 45% of the U.S. population has Type O blood, 40%
Type A, 11% Type B and the rest Type AB.
a) Someone volunteers to give blood. What is the probability that this donor
1. Has Type AB blood?
2. Has Type A or Type B?
3. Is not Type O?
b) Among four potential donors, what is the probability that
1. All are Type O?
2. No one is Type AB?
3. They are not all Type A?
4. At least one person is Type B?
What does your GPA say?
Five hundred first-year students at a state university were classified both according to high
school GPA and whether or not they were on academic probation at the end of their first
semester (adapted from Devore & Peck, p. 182)
Probation [2.5, 3.0) [3.0, 3.5) 3.5+ Total
Yes
50
55
30
135
No
45
135
185 365
Total
95
190
215 500
A) What is the probability that a randomly selected first-year student at this university is on
academic probation after their first semester?
B) What is the probability that a randomly selected first-year student at this university is on
academic probation and has a high school GPA of 3.5 or better?
C) What is the probability that a randomly selected first-year student at this university is on
academic probation if they had a high school GPA of 3.5 or better?
D) Are “having a GPA of 3.5+” and “being on academic probation” independent events?
AP Scores
Given the following distribution of AP Scores:
AP Score
P(score)
1
.05
2
.1
3
.4
4
.2
(P is the probability of getting a certain AP Score)
a) Find the expected value of AP Score
b) Find the variance and standard deviation of the AP Scores.
Kittens (eeeewwww) (Book BVD p. 383 #22)
In a litter of seven kittens, three are female. You pick two kittens at random.
a) Create a probability model for the number of male kittens you get.
b) What’s the expected number of males?
c) What’s the standard deviation?
5
.25
Ostrich Eggs
A large quantity of ostrich eggs are placed in incubators at different temperatures. The
probabilities of the eggs hatching are given by the following distribution:
Temperature
(in F)
P (x)
99
100
101
102
103
.2
.15
.3
.3
.05
a) Find the expected value of the temperature in degrees F.
b) Find the variance and standard deviation of the temperature in degrees F.
c) Find the expected value of the temperature in degrees C.
d) Find the variance and standard deviation of the temperature in degrees C.
Test
The last test had a mean of 81.2 and a standard deviation of 4.5. Suppose I decided to be
“creative” with your curve and chose to do this:
New Grade = ¾ Original grade + 25.
What would the mean and standard deviation of this new distribution be?
Assembly Line
In an assembly line process, the mean time it takes to position a part to be attached to an
automobile chassis is 5.1 seconds with a standard deviation of 1.1 seconds. The mean time it
takes to attach the part is 4.2 seconds with a standard deviation of .7 seconds. If we are interested
in a distribution of combined time to position the part and attach it, what would the mean and
standard deviation of that distribution be?
A Very Special Day
On a very special day, a store offers 100 distinct items at a discount of 20% Each item is subject
to a 6% sales tax.
Prior to applying the discount and sales tax, the mean of the 100 prices is $200 and the standard
deviation is $10,
1) If the discount is applied BEFORE the sales tax, what are the mean and standard deviation of
the 100 selling prices on the special day?
2) If the discount is applied after the sales tax. What are the mean and standard deviation of the
100 selling prices on the special day?