7.1 How
Probabilities are determined
|
Roll a die |
1,2,3,4,5,6 |
{1,2,3,4,5,6} |
Flip a Coin
Experimental (Relative Frequency) vs. Theoretical Probability
Event A
particular subset of the Sample Space
DISCUSS various
EVENTS from flipping two coins
|
Sample
Space = A = one
of each |
|
|
B = at
least 1 head |
|
|
C = both
tails |
|
In terms of
probability flipping 2 coins is equivalent to having two children and
considering boy versus girl possibilities.
Sample
Space for a family with two children
Experiment
of 3 kids in a family
Sample
Space =
List some
possible Events
Determining
Probability
Empirical
versus Experimental Probability
Empirical -
theoretical – relative frequency
Experimental -
observe many times and record to get relative
frequency
Theoretical Probability
Roll a die P({5}) = 1/6
Uniform Sample
Space - An experiment with equally likely
outcomes
Definition of
Probability of an Event with Equally Likely Outcomes
For an
experiment with sample space S and equally likely outcomes, the probability
of an event a is given by:
Range of
Probability Values
0 < P(A) < 1
Impossible Certain
Property for Probability of an Event
The
probability of an event is equal to the sum of the probabilities of the
disjoint outcomes making up the event.
Example
S = {1,2, …, 15} Chose a
number at random
|
Find
Probability of Event |
Event |
Description
of Event |
|
|
A |
Odd
Number drawn |
|
|
B |
Number
>5 and <12 |
|
|
P |
Prime
Number drawn |
|
|
E |
Even
Number drawn |
|
|
|
Number is
Even OR Prime |
|
|
|
Number is
Even AND Prime |
Mutually
Exclusive Events
Events A
and B are mutually exclusive if they have no elements in common – that
is 
Property of Probability of Union of
DISJOINT Events
If events A
and B are mutually exclusive then
Spin a die
Flip two
coins: A = Both Heads B = Both Tails
Complementary
Events
If A is an event and 
is its complement,
then
Probability
of rain is .3, what is probability that it will not rain?
If roll two die, what is probability that do not get doubles?
If roll two die, what is probability get a sum of 4 or more?
In a family
of 4 kids, what is probability that they are not all boys or all girls?
Birthday
Problem
In a group of ____,
what is the probability that at least two people have the same birthday.
Events
which are Not Mutually Exclusive
sets with non-empty intersection
Example
S= {1, 2, 3, …, 15} E =
multiples of two and F = multiples of
three
Find 
Find 
Probability of non-Mutually
Exclusive Events
Summary of
Properties of Probability
Sample Space for
Roll Two Die - Page 442
{(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)}
Consider
several probabilities:
Probabilities
for Sum of faces of two die
Experiment
– Marbles in a bag – 3 Red, 2 green, 5 Black … draw 1
P(R)
P(not R)
P(G or Bl)
Be VERY CAREFUL of Not Equally likely - some spinners, or weird die
Cards –
Deck of cards – draw 1
P(Ace)
P(Heart)
P(Heart
& Ace)
P(Heart
or Ace)