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Prob and Stats, Oct 9
Probability I – Introduction and the Definitions
in Probability, Simple Probability
Book Sections: N/A
Essential Questions: How can I compute the probability of any event?
What is probability and what is the mathematical basis of
computing it? How do I compute simple probability?
Standards: PS.SPCR.1, .1a, .1b
Probability
What is probability?
• A measure of the likelihood that an event will
happen.
Simple Probability
Terms to Know and Understand
• Outcome – The result of a random phenomenon
• Random –outcomes that occur by chance, and
every outcome has the same chance or is equally
likely
• Event – A collection of outcomes
• Probability – The likelihood that an event will
happen
• Sample Space – A list of all possible outcomes
of an event
Notation
• The probability of an event will be
abbreviated as follows:
P(event) =
Examples
In a coin flip there are two possible outcomes,
head and tail
The probability of the outcome head of the event
would be expressed as: P(head)
The probability of the outcome tail of the event
would be expressed as: P(tail)
More Examples
The Mathematical Basis of Probability
• You compute probability, no matter how it
is expressed, by using a ratio.
• A ratio is comparing two numbers by
division.
The most common manifestation of a ratio is a
fraction.
Ratio
• A ratio is a comparison of two numbers by
division
• There are three ways of writing a ratio:
5
5 to 22, or 5:22, or
22
Examples
1) 15 cats to 50 dogs
2) 27 nurses to 9 doctors
3) 8 completions:12 passes
4) 21 hired out of 105 applicants
5) 2 teams : 22 players
6) 7 first downs to 28 first downs
Expressing Probability
• Probability can be expressed as:
A fraction (in simplest form)
A decimal
A percent (%)
• Probability ranges between 0 and 1 (0% - 100%)
Probability of 0 means the event is impossible
Probability of 1 means the event is a sure thing
The Range of Probabilities
likely
What are Favorable Outcomes?
• Favorable outcomes of some event A are
the ways within the sample space that A can
happen.
Examples
Probability in this Course
• We will compute simple and every other type
of probability you can think of in this class.
You will need to know how to: Create and
simplify fractions, add fractions, multiply
fractions over the course of this unit.
• The calculator can do all of these, understand
how and know the calculator limitations!
The Mathematical Definition of Probability
Number of favorable outcomes
P(event) =
Total number of outcomes
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.
What Are the Components of the Ratio
• Favorable outcomes – The number of ways within the sample space
that the event (what you want to occur) CAN occur
• Total number of outcomes – everything that can happen in the
sample space
• In these two counts you are counting the options, the outcomes are
not the options – it is the number of outcomes
The total number of outcomes is also known as all possible
outcomes
Some Simple Examples
Favorable Outcomes (How Many?)
If you:
Select the number 5 on a roll of a die. = 1
Select an even number on a roll of a die. = 3
Select a red card from a deck of cards. = 26
Select a queen from a deck of cards. = 4
Select the 5 of diamonds from a deck of cards. = 1
Call heads on a coin flip. = 1
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 those are all the numbers on the standard cube
Drawing a single card from a deck. There are 52 possible outcomes,
because there are that many cards in the deck.
Flipping a coin has two possible outcomes, a head or a tail, there are
two sides on every coin.
Computing Probabilities
The probability of the number 5 on a roll of a die.
The probability of an even number on a roll of a die.
The probability of a red card from a deck of cards.
The probability of a queen from a deck of cards.
The probability of the 5 of diamonds from a deck of cards.
The probability of heads on a coin flip.
Examples
P(4) =
P(red) =
P(even) =
P(odd) =
P(not 5) =
P(9) =
P(4 or 5) =
P(< 8) =
Examples
P(4) =
P(red) =
P(even) =
P(odd) =
P(not 5) =
P(blue) =
P(4 or 5) =
P(prime) =
Examples
A bag contains 3 pink, 2 blue, 5 black, 1 clear and 1 yellow
marble. Compute the following probabilities based on The
probability a single marble from the bag:
P(black)
P(yellow)
P(green)
P(clear)
P(not blue)
P(yellow or blue)
Is it Really Random? Or Not
When is an occurrence random?
Answer: When every outcome is equally likely.
What About These?
Any problems here?
Complementary Events
The compliment of an event A is everything
happening except A
The compliment of A is called not A and is
abbreviated with A (called as A bar) or not A
Examples
The compliment of heads on a coin flip would be not heads which would be ‘tails’
The compliment of a 5 on a dice roll (not 5) - would be 1, 2, 3,
4, or 6
The compliment of a rainy day forecast for today - would be no
rain today
Complementary Events
An event will either happen or it will not.
All possible outcomes that are not an event add up to be the
compliment of that event.
The sum of the probability of an event and the event’s
compliment always add up to 1 or
P(event) + P(not the event) = 1, where P(not the event) is
the probability of the event’s compliment.
In Other Words
• The probability of a complimentary event is:
P(event) = 1 – P(event) or P(A) = 1 – P(A)
Example: What is the probability of not getting a
3 when rolling a die?
1 5
1 – P(3) = 1  
6
6
Examples
Compute each probability
P(not even)
P(red)
P(not yellow)
If the P(odin) = ⅜, what is P(not odin)?
Class work: CW 10/9/15, Parts 1&2, All
Homework: None