Chapter 5 Required Learning Outcomes - Probability
The student will be able to:
- Find probabilities of simple events.
- Name the sample space.
- Compute probabilities using the addition rule.
- Compute probabilities using the multiplication rule and the general multiplication rule.
- Use the complement rule.
- Calculate conditional probabilities with contingency tables.
- Compute and interpret empirical and theoretical probabilities using the rules of probabilities and
combinatorics.
- Use probability to determine if an event is unusual.
- Use a graphing calculator to find various probabilities.
Chapter 5 Formulas
Section 5.1 –Probability Rules
Objectives
- Apply the rules of probabilities
- Compute and interpret probabilities using the empirical method
- Compute and interpret probabilities using the classical method
- Use simulation to obtain data based on probabilities
- Recognize and interpret subjective probabilities
Probability deals with experiments that yield short-term results and reveal long-term predictability, i.e., how likely it is that some event will happen.
The long-term proportion with which a certain outcome is observed is the probability of that outcome.
Probability experiment – Any process with uncertain results that can be repeated. From Wikipedia: any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space.
Law of Large Numbers–As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome.
Sample Space – The collection of all possible outcomes. Denoted S.
Event – A collection of outcomes from a probability experiment. (may consist of one or more outcomes)
Events are usually denoted with capital letters.
Example 1
Roll a single fair die and note the number of dots showing face up. What is the sample space? Define the event E as “rolling an odd number.” List E.
Example 2
Flip a coin three times. What is the sample space?
P(E) means the “probability that event E occurs”
Rules for Probabilities:
- For any event E, 0 ≤ P(E) ≤ 1.
- The sum of the probabilities of all outcomes must equal 1
Probability rule #2 looks like this:
If S = {e1, e2, … , en} then P(e1) + P(e2) + … + P(en) = 1
Definitions:
- An impossible event has probability 0
- An event certain to occur has probability 1
- An event with a probability less than 0.05 (5%) is considered unusual. However, 0.05 is subjective and in some instances may change to 0.01 or 0.10.
A probability model lists the possible outcomes of a probability experiment along with each outcome’s probability. A probability model must satisfy Probability Rules 1 and 2 above.
Example: The following probability model shows the colors of M&M’s along with their probabilities.
Example: Is the following a probability model?
Two Ways to Compute Probability:
- Empirical Approach
- Classical Approach
Empirical Approach
Repeat an experiment a number of times and record how many times event E occurs.
P(E) relative frequency of E =
The more times the experiment is repeated the more accurate the result (Law of Large Numbers)
Example: Suppose a survey asked 500 families with three children to disclose the gender of their children and found that 180 of the families had two boys and one girl. Estimate the approximate probability of having a three-child family with two boys and one girl using the empirical approach.
Classical Approach
Each simple event must have an equal chance of occurring (i.e., outcomes must be equally likely)
P(E) =
Example: Compute the probability of having a three-child family with two boys and one girl using the classical approach.
Example: Roll a pair of fair dice and count the number of dots showing face up.
S = {
(1,1) / (1,2) / (1,3) / (1,4) / (1,5) / (1,6)(2,1) / (2,2) / (2,3) / (2,4) / (2,5) / (2,6)
(3,1) / (3,2) / (3,3) / (3,4) / (3,5) / (3,6)
(4,1) / (4,2) / (4,3) / (4,4) / (4,5) / (4,6)
(5,1) / (5,2) / (5,3) / (5,4) / (5,5) / (5,6)
(6,1) / (6,2) / (6,3) / (6,4) / (6,5) / (6,6)
}
What is the probability of getting a sum of 9?
What is the probability of getting a sum of 7?
What is the probability of getting sum less than 5?
What is the probability that one of the die is a 6?
Example: A roulette wheel has 38 slots (0, 00, 1–36). You bet on an even number.
a) P(win) =
b) P(lose) =
Example: Flip a coin 2 times. What is the probability that tails comes up both times?
Section 5.2 – The Addition Rule and Complements
Objectives
Use the Addition Rule for Disjoint Events
Use the General Addition Rule
Compute the probability of an event using the Complement Rule
Disjoint events - Events that have no outcomes in common. Also called mutually exclusive.
Venn Diagram:
Represents events as circles enclosed in a rectangle representing the Sample Space.
Addition Rule for Disjoint Events:
If E and F are disjoint events, then P(E or F) = P(E) + P(F)
Example: Roll a pair of fair dice and count the number of dots showing face up.
P(sum is 4 OR sum is 10)
P(sum is 7 or less OR sum is more than 7)
Example: A single card is drawn from a deck of cards. Calculate the following probabilities:
a) P(draw a 5) =
b) P(draw a 5 OR a face card) =
c) P(draw a 5 OR a face cardORan Ace) =
Example: A bag contains 5 red marbles, 3 blue marbles, and 2 yellow marbles. You reach into the bag and select one marble. Calculate the following probabilities:
a) P(marble is red OR blue) =
b) P(marble is blue OR yellow) =
More Venn Diagrams:
INTERSECTION: P(E and F) UNION: P(E or F)
P(E and F) = P(E and F both occur)P(E or F) = P(E occurs or F occurs)
P(E and F) is also written as
P(E or F) is also writer as
The General Addition Rule:
P(E or F) = P(E) + P(F) – P(E and F)
Important: make sure that no outcome is counted more than once!
Example: A single card is drawn from a deck of cards. Calculate the following probabilities:
a) P(King OR Red) =
b) P(Diamond OR face card) =
Example: Roll a pair of fair dice and count the number of dots showing face up. What is P(rolling an even number or a number less than 7)?
The Complement of E – All outcomes in the Sample Space that are not in E. Denoted .
Complement Rule:P() = 1 – P(E)
Note that together E and Ecmake up the sample space.
Example: Roll a pair of fair dice and count the number of dots showing up. What is the probability of rolling a number greater than 2?
Finding Probabilities Using a Contingency Table:
A contingency table relates two categories of data.
Probability that a randomly selected fatal crash involved a male = P(male) =
Probability that a randomly selected fatal crash occurred on Sunday = P(Sunday) =
P(male and Sunday) =
P(male or Sunday) =
Is P(female and Wednesday) and unusual event?
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