Download Statistics Introduction to Probability Unit Plan

Statistics Introduction to Probability Unit Plan
Unit Key Idea: Probability is the attempt to describe ideas like “chance” and “likelihood” in a
mathematically precise way.
I. Basic Probability
Key Idea: There are two ways to think about probability: Theoretical and Experimental.
Key Idea: Probabilities can be expressed as fractions, decimals or Percents.
Definitions:
Trial- Any action or set of actions with an uncertain outcome.
Ex: Rolling a pair of dice vs. flipping a light switch
Sample space – The set of all possible outcomes of a trial.
Random – A sample space is said to be random if all outcomes in it are equally likely. This is what it means
when a problem says things are “chosen at random” or “chosen randomly.”
Event – Any subset of a sample space. Generic symbol: “E”
(Ask the class if they know what a subset is.)
Events can either be simple, when they consist of a single outcome, or compound, when they consist of
more than one outcome.
There are two definitions of probability…
Theoretical Probability (of an Event “E”) – Symbol: P(E). The number of outcomes in E divided by the
total number of outcomes in the sample space:
𝑃(𝐸) =
|𝐸|
⁄|Ω|
NOTE: This definition only works if all outcomes in a sample space are equally likely (random).
The size of any sample space can be found by the Fundamental Counting Principle: If there are “m”
possible results for the first action in a trial, and “n” possible results for the second action in a trial, the
total number of outcomes in the sample space is 𝑚 ∙ 𝑛 . This principle can be extended for any
number of actions in a trial.
A probability experiment is a statistical study that consists of conducting the same trial multiple times and
recording the result each time. The number of trials conducted is equivalent to a sample size, so it is given
the same variable, 𝑛 .
Conducting a probability experiment reveals the other way to think about probability:
Experimental Probability (of an Event “E”) –The number of trials in which an outcome in occurred divided
by the total number of trials (n) .
𝑃(𝐸) =
𝑛𝐸⁄
𝑛
A consequence of these definitions is that the probability of any Event is always between zero and 1. We
call this the range rule: 0 ≤ 𝑃(𝐸) ≤ 1
What links these two concepts of probability together?
Law of Large Numbers – As the number of trials increases, experimental probability tends to converge to
theoretical probability, as long as the theory is correct and the data collection is unbiased.
There is a “third” kind of probability, based on an initial estimate from intuition, not supported by any
numerical evidence. This is called subjective probability.
Subjective Probability – A guess. This guess can be refined with collecting data as we will see in a later
section.
Ex: Sam the Skeptic thinks it is very unlikely that ghosts exist. He estimates the probability that they exist
at 1%.
Complement- Symbol: E’ (“E prime”.) All the outcomes in the sample space that are not in Event E.
P(E’) = 1 – P ( E )
Activity: Probability Experiment Worksheet (Quarter activity)
HW: Basic Probability Topic Practice
II. Conditional Probability
A conditional probability is the probability that an outcome is in an Event given that it is in another Event.
For example, the probability that a randomly selected person is over 6 feet tall is much less than the
probability that a randomly selected person is over 6 feet tall given that they are an NBA player.
The conditional probability of an outcome being in Event B given that it is in Event A is found using the
formula:
𝑃(𝐵|𝐴) = (# 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑡ℎ𝑎𝑡 𝑎𝑟𝑒 𝑖𝑛 𝑏𝑜𝑡ℎ 𝐴 𝑎𝑛𝑑 𝐵)/(# 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝐴)
This formula is theoretical and only works if all outcomes are equally likely to occur. Notice that the
denominator is the # of outcomes in A, not in the whole sample space.
The line found on the left-hand side is the symbol for “given that.” The left-hand side of the formula is read
“The probability of B given A.”
Conditional probability problems can often be visualized using a Venn Diagram.
Ex: Collect some data from the class that can be divided into a sample space with “just A, just B, both and
neither” regions. Possible topics: Do you play a sport and do you have a job? Do you have a cat or a dog? Do
you have a brother or a sister?
An important idea associated with conditional probability is independence. Two events A and B are said to
be independent of each other if and only if:
𝑃(𝐴) = 𝑃(𝐴|𝐵) 𝑜𝑟 𝑃(𝐵) = 𝑃(𝐵|𝐴)
These two equations are equivalent statements. That is, if the one is true, the other is true, and if the one
is false, the other is false. Therefore to check if two Events are independent of one another, you only need
to check one of the equations (whichever one is more convenient for you.)
Events that are not independent of one another are said to be dependent on one another.
Ex: Check the independence relationship between the Events in the Venn Diagram.
HW: Conditional Probability Topic Practice
III. Compound Probability
In this section we will learn how to find the probability that an outcome is in more than one event (“and”
probabilities), and the probability that an outcome is in at least one event (“or” probabilities.)
Key Idea: “And” means multiply; “Or” means add (and sometimes subtract)
Definitions:
Disjoint (aka Mutually Exclusive) – Two events are called disjoint if they have no outcomes in common.
Overlapping – Two events are called overlapping if they have at least one outcome in common.
Multiplication Rule: P(A and B) = P(A)*P(B|A). Recall that the “|” means “given that.”
Example: 40% of students in a room are seniors. 25% of the seniors are football players. If you choose a
student at random from this room, what is the probability that they are a Senior and a football player?
Events in Sequence
Events in sequence are “and” probabilities, because saying “I flipped a coin, then rolled a die” could be
re-written as “I flipped a coin and then I rolled a die.”
Ex: A trial consists of flipping a coin and then rolling a die. What is the probability that the outcome of
this trial was (Heads, 5) ?
Ex: A box contains five white, three red and two blue marbles. A trial consists of picking a marble at
random, putting it back, and then picking another marble at random. What is the probability that both
the marbles picked are red?
What is the probability that both the marbles picked are red if you don’t put the first marble back
before picking the second one?
Key Idea: Events in sequence with replacement are independent of each other. Events in sequence
without replacement are dependent.
Ex: I flip a fair coin ten times in a row. What is the probability that it came up heads every time?
Key Idea: Independent Events in sequence can be found using the power principle. It states that if “r”
independent Events in sequence all have the same basic probability “p,” then
𝑃(𝐸1 𝑎𝑛𝑑 𝐸2 𝑎𝑛𝑑 … . . 𝐸𝑟 ) = 𝑝𝑟
Addition Rule: P(A or B) = P(A) + P(B) – P(A and B). The subtraction term is necessary when A and B are
overlapping, because we counted events in the “middle” twice.
Example: In a classroom of 30 students, 20 play sports and 15 are in the band. 8 sports-players are also
in the band. What is the probability that, if you picked a student at random from this room, he or she
would play sports or be in the band? [Answer: 27/30 = 90%]
HW: Compound Probability Topic Practice;
Optional HW: p. 134-138 #1-26; p. 145-148 #1-25 (hand out evens answer key if this is assigned.)
Basic, Conditional and Compound Probability Quiz
IV. Random Variables
A random variable is defined by a rule that assigns a number to every outcome in a sample space.
Generically it is given the letter 𝑥 .
The set of all outcomes assigned to a single number is called an instance.
Ex: Consider a trial that consists of flipping two coins. First of all, what are the possible outcomes?
Ω = { (𝐻, 𝐻); (𝐻, 𝑇); (𝑇, 𝐻); (𝑇, 𝑇)}
Let 𝑥 = the number of heads seen in the trial. How many instances of 𝑥 exist? [Answer: 3 instances: 0,
1, and 2.]
To find the (theoretical) probability of any random variable, use the formula:
𝑃(𝑥) =
|𝑥|
|Ω|
where |𝑥| is the number of outcomes in any instance of 𝑥
𝒙
0
1
2
∑
Outcome(s)
(T, T)
(H, T) and (T, T)
(H , H)
𝑃(𝑥)
1/4 = 0.25
2/4 = 0.50
1/4=0.25
1
An important rule associated with random variables is the sum rule: ∑ 𝑃(𝑥) = 1 . This can be used to
check our work!
Diagramming Sample Spaces to Calculate Random Variables
There are a number of methods to diagram the outcomes in a sample space and assign numbers to its
outcomes. In this section we will be looking at two: The probability tree and the set of axes. In later
sections we will see other ways to diagram sample spaces.
a.) Set of axes
These are useful for trials that consist of two actions, such as flipping two coins or rolling a pair of
dice. Consider a trial that consists of rolling two dice and then let 𝑥 = the sum of the numbers
that appear on the dice. How would we diagram this sample space?
[Create the dice diagram and a probability table for 𝑥 .]
b.) Probability trees
These are useful for trials that consist of more than two actions, such as flipping three coins. Each
result is represented by a “node,” and each action by a “branch.” The outcomes at the ends are the
“leaves” of the tree. Let’s see what this means:
Consider a trial that consists of flipping three coins. Let 𝑥 = the number of heads that appear in
this trial.
[Diagram a tree for flipping three coins and create a probability table for 𝑥]
HW: Random Variables Topic Practice
V. Non-random Sample Spaces
Probability trees are also useful for diagramming sample spaces that are not random, that is, where all
outcomes are not equally likely.
[Show football example.]
Another approach to this is to re-define the number of outcomes so that the sample space is random. This
is called a synthetic sample.
[Show the football example re-cast as a set of axes problem.]
HW: Weighted Probability Trees Topic Practice
Probability Tables Topic Practice
VI. Odds
Odds are an alternate way to express likelihood. Odds are not probabilities, but they express similar ideas.
There are actually two types of odds: Odds in favor and odds against.
The odds in favor of an event E are written in the form |𝐸|: |𝐸 ′ | , where 𝐸′ is the complement.
The odds against an event E are just the reverse: |𝐸 ′ |: |𝐸| .
Ex: A sock drawer contains 3 red, 4 white and 5 blue socks. If a trial consists of picking a sock at random
from this drawer, what are the odds in favor of a white sock?
4: 8 = 1: 2
Key Idea: Though odds are not fractions, they do reduce like fractions.
Ex: A sock drawer contains 3 red, 4 white and 5 blue socks. If a trial consists of picking a sock at random
from this drawer, what are the odds against a red sock?
9: 3 = 3: 1
Converting odds to probabilities (and vice versa)
Ex: A horse has a 30% chance of winning a race. What are the odds against that horse winning?
Since there is a 30% chance the horse will win, there is a 70% chance that it will lose, so the odds against
are:
70: 30 = 7: 3
Ex: The odds in favor of winning a prize in the raffle are 1: 12 . What is the probability of winning a prize?
Since there is one way to win for every 12 ways to lose, we can see this as a problem where there are 13
total outcomes in the sample space, so
𝑃(𝑊𝑖𝑛) =
1
13
HW: Odds Topic Practice; p. 129 #41-44 (these are odds in favor) if necessary
VII. Unit Review
Introduction to Probability Worksheets #1-4
Introduction to Probability Study Guide
Introduction to Probability Test