Geometry HSAP Review, Mar 26
Probability
Book Sections: N/A
Essential Questions: What is simple probability, what are its components, where do I find it and how do I compute it? What is
multiple event probability?
Standards: N/A
Part 1
• Simple probability – One thing happening.
Notation
• The probability of an event will be
abbreviated as follows:
P(event) =
The Mathematical Definition of Probability
P(event) =
In words: The probability of an event is the ratio of favorable outcomes to the number of possible outcomes. That number will always be between 0 and 1.
Number of favorable outcomes
Total number of outcomes
What Are the Components of the Ratio
• Favorable outcomes – The number of ways within the sample space
that what you want to occur CAN occur
• Total number of outcomes – everything that can happen in the
sample space
The total number of outcomes is also known as all possible
outcomes
Some Simple Examples Total Number of Outcomes
If you roll a single fair die, there are 6 possible outcomes, which are
1, 2, 3, 4, 5, 6
Drawing a single card from a deck. There are 52 possible outcomes.
Flipping a coin has two possible outcomes, a head or a tail.
Some Simple Examples Favorable Outcomes
Betting on the number 5 on a roll of a die.
Selecting an even number on a roll of a die.
Selecting a red card from a deck of cards.
Selecting a queen from a deck of cards.
Selecting the 5 of diamonds from a deck of cards.
Calling heads on a coin flip.
Computing Probabilities
The number 5 on a roll of a die.
Selecting an even number on a roll of a die.
Selecting a red card from a deck of cards.
Selecting a queen from a deck of cards.
Selecting the 5 of diamonds from a deck of cards.
Calling heads on a coin flip.
More Examples
Expressing Probability
• Probability can be expressed as:
A fraction (in simplest form)
A decimal
A percent (%)
• Probability ranges between 0 and 1
Probability of 0 means the event is impossible
Probability of 1 means the event is a sure thing
Examples A bag contains 4 red, 3 blue, 2 green, and 1 yellow
marble. Compute the following probabilities based on
selecting a single marble from the bag:
Part 2
• Multiple event probability – More than one
thing happening.
The Probability of a Single Event
P(event) =
Number of favorable outcomes
Number of possible outcomes
Multiple Event Methodology
• Anytime we do multiple events, and they
involve the word and – meaning both, we
are multiplying probabilities.
Multiple Events
• Multiple Events – More than one random
event occurring simultaneously or in
succession.
• Multiple events are either independent or
dependent events
How Can I Tell the Difference
• Drawing or selecting anything from a pool
of objects and then selecting another
without replacing the first is an indication of
dependency
• A situation where there is a cause – effect
relationship
• Anything else, including selection with
replacement or from different sources, are
independent events
Independent or Dependent?
1. Tossing a coin and spinning a spinner
2. Drawing two cards from a single deck
3. Drawing two cards from separate decks
4. Selecting two marbles from a bag of marbles
5. Being a lifeguard and getting a suntan
6. Betting on different horses to win, place, and
show at the track
7. Rolling two dice
8. Parking in a no parking zone and getting a
parking ticket
I
D
I
D
D
D
I
D
Multiple Event Probabilities 1
• Multiple events are called event A and event B
A and B are independent
• P(A and B) = P(A) · P(B), P(A) and P(B) are simple
probabilities
An Example
P(7and 4)
P(even and odd)
P(yellow and red)
Multiple Event Probabilities 2
• Multiple events are called event A and event B
A and B are dependent
• P(A and B) = P(A) · P(B | A), where P(A) is a simple
probability and P(B | A) is a probability computed for B
given that A has already happened
Example • Bag A contains 10 marbles of the following colors:
4 green, 3 blue, 2 red, and 1 purple.
P(blue and red)
P(red and green)
P(green and gray)
P(green and green)
Adjusting a Sample Space
To compute P(B following A) – Two Steps to
adjusting a sample space:
1. There are now one less items in the space
2. The item ‘selected’ is not there, debit that
mini pool.
Multiple Event Probabilities
• Multiple events are called event A and event B
A and B are dependent
• P(A and B) = P(A) · P(B | A), where P(A) is a simple
probability and P(B | A) is a probability computed for B
given that A has already happened
A and B are independent
• P(A and B) = P(A) · P(B), P(A) and P(B) are simple
probabilities
Multiple - Multiple Events
• What if its more than two events?
• Assess the situation, compute probabilities,
keep multiplying
If dependent, think P(C | A and B)
Example
A person owns a collection of 30 CDs, of which 5 are country
music. If two CDs are selected randomly, what is the
probability that both are country music?
Working With All Multiple
Events
1. Determine dependency (Indep or Dep)
2. Visualize or draw a picture of sample space
3. Compute probability of first event
4. Compute probability of second event
Adjust sample space if a dependent event
5. Multiply probabilities together
6. Simplify (if required)
Homework: None
Class work: Handout CW 3/26, all