Download Intro to Probability

Intro to Probability
Objectives:
To evaluate the big idea in Probability:
chance behavior is unpredictable in the
short run but has a regular and
predictable pattern in the long run
(Law of Large Numbers).
To use Venn diagrams as a visual aid
for understanding concepts in context
Warm-up:
1. If you have 6 different books to place on a
bookshelf, how many different arrangements
are possible?
654321=720
2. How many different combinations exist for
your locker?
503 =125000
3. How many different locker combinations
would there be if you could only use each
number once? 504948 =117600
Questions in modern day probability:
 Should I spend money on a warranty for
my new Ipod?
 If I test positive for a rare blood disease,
does this mean that I definitely have this
disease?
 Can we determine the chances of a child
having a psychological disorder based on
heredity?
Random Phenomenon
A phenomenon is
random if individual
outcomes are uncertain,
but there is a predictable
distribution of outcomes
over many repetitions.
Experimental versus Theoretical Probability
Probability
The probability of any outcome of a
random phenomenon is the
proportion of times the outcome
would occur in a very long series of
repetitions. (Probability is basically
long term relative frequency)
Sample Space (S)
The set of all possible outcomes for some
type of random phenomenon
Examples:
 Coin Toss S = {H, T}
 Fair die
S = {1, 2, 3, 4, 5, 6}
 Toss a coin twice S = {HH, TT, HT, TH}
How many outcomes are there in the sample
space for rolling two dice?
36
Event
An event is any outcome or a set of outcomes of
a random phenomenon.
An event is basically a subset of the sample
space.
Examples:
Rolling a Prime #
A = {2, 3, 5}
Rolling a Prime # or an even #
B = {2, 3, 4, 5, 6}
Complement
c
E
 Consists of all outcomes that are not in the
event
Example:
Rolling an even #
E={2,4,6}
Complement: Not rolling an even #
EC={1,3,5}
Union
E  AB
the event A or B happening
consists of all outcomes that
are in at least one of the two
events
Ex. Rolling a prime # or even number
W ={2,3,4,5,6}
Intersection E  A  B
the event A and B happening
consists of all outcomes that
are in both events
Example:
Drawing a red card and a “2”
L = {2 of hearts, 2 of diamonds}
Mutually Exclusive (disjoint)
two events have no outcomes in
common
Example:
The event of rolling an even # is
disjoint from the event of rolling an
odd #
Probability model
Mathematical description of a random
phenomenon consisting of two parts
1. A sample space (S)
2. A method of assigning probabilities to
each event
We will focus on part 1 today…
Tree Diagram
It is very important to check that we have
not overlooked any possible outcome.
One visual method of checking
this is making use of a
tree diagram.
Ex. Flip a coin, then roll a die
S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
Multiplication Principle
If you can do one task in x number of ways and a
second task in y number of ways, then the sample
space of both task can be shown with x ● y
possible outcomes.
Example:
Tossing a coin – two possible outcomes
Rolling a die – six possible outcomes
Tossing a coin, then rolling a die:
2 ● 6 = 12 possible outcomes
Venn Diagrams
Used to display
relationships between
events
Helpful in calculating
probabilities
Venn diagram - Complement of A
AA
Venn diagram - A or B
A
B
Venn diagram - A and B
A
B
Venn diagram - disjoint events
A
B
Independence
 The outcome of one trial must not influence
the outcome of another trial.
 This is a major concept in statistics that is
often neglected in the design and data
collection process.
 We will look at independence both logically
and mathematically in this course.
Closing:
1. In your own words, describe random phenomena.
1. Write down the symbol, key word, and visual display
for Union
1. Write down the symbol, key word, and visual display
for Intersection