Download 6.5 Combining Probabilities

Statistical Reasoning
for everyday life
Intro to Probability and
Statistics
Mr. Spering – Room 113
6.5 Combining Probabilities
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Find the probability.
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What is the probability of rolling 2 or a 5 on a number cube?
2/6 or 33.33%
A bag contains 32 red marbles, 30 blue marbles, and 18 white
marbles. You pick one marble from the bag. Find P (picking
blue).
3/8 or 37.5%
P (not a red)
3/5 or 60%
What is the probability of having a sample with mean age
between 35 years and 45 years, given the population mean is
40 years and the standard of deviation is 2.5 years?
95%
Using a regulation deck of cards. What is the probability of
choosing a Queen of Hearts?
1/52, 0.019, or 1.9%
6.5 Combining Probabilities
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Contingency Tables
Ace
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Tree Diagrams
Sample
Space
Full Deck
of 52 Cards
Not Ace
Total
Black
2
24
26
Red
2
24
26
Total
4
48
52
2
24
2
24
Sample
Space
6.5 Combining Probabilities
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Venn Diagrams
 Let A =
 Let
aces
B = red cards
A ∩ B = ace and red
A
A U B = ace or red
B
6.5 Combining Probabilities
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PERMUTATIONS = Arrangements (Order matters)
Permutations: The number of ways of arranging X objects
selected from n objects in order is
n!
n Px 
(n  X)!

Example:
 Your restaurant has five menu choices, and three are selected
for daily specials. How many different ways can the specials
menu be ordered?

Answer:
nPx 
n!
5!
120


 60
(n  X)! (5  3)!
2
different possibilities
6.5 Combining Probabilities
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COMBINATIONS = Grouping (Order does not matter)
Combinations: The number of ways of selecting X
objects from n objects, irrespective of order, is

n!
n Cx 
X!(n  X)!
Example:
 Your restaurant has five menu choices, and three are
selected for daily specials. How many different special
combinations are there, ignoring the order in which they are
selected?
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Answer:
n!
5!
120


 10
n Cx 
X!(n  X)! 3! (5  3)! (6)(2)
different possibilities
6.5 Combining Probabilities
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Joint Probabilities (AND probabilities)
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Independent VS. Dependent…
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Independent events are events where the outcomes of one does not
affect the outcomes of another. Dependent events are events where
the outcome of one will affect the outcome of another.
Independent → flipping a coin
Dependent → Drawing two cards after drawing a card
6.5 Combining Probabilities
 Independent…
 AND probability…

Considering two independent events A and B
that have individual probabilities P(A) and P(B).
The probability that A and B occur together is:
P( A and B)  P( A)  P( B)
 Concept
may be extended for more than 2 events.
P( A and B and C )  P( A)  P( B)  P(C )
6.5 Combining Probabilities
 Independent…Example

Suppose you have a coin and a spinner with 5
equal sectors, labeled 1 thru 5. What is the
probability of spinning an even number AND
getting heads?
P( A and B)  P( A)  P( B)
1 2 2 1
P(heads and even)   

2 5 10 5
6.5 Combining Probabilities
 Dependent
…{Conditional Probability}
 AND probability…

Considering two events A and B. The
probability that A and B occur together is:
P( A and B)  P( A)  P( B given A)
 Concept
may be extended for more than 2 events.
P( A and B and C )  P( A)  P( B given A)  P(C given A and B)
6.5 Combining Probabilities
 Dependent

…
The game of BINGO involves drawing pieces with a letter
and a number on each piece. If we draw at random
without replacement. Find the probability of drawing two
B pieces in the first two selections, given there are 75
pieces, 15 for each of the letters B, I, N, G, O!
P( A and B)  P( A)  P( B given A)
15 14
7
P(B and B) 


 0.0378
75 74 185
6.5 Combining Probabilities
•Either/OR Probability: [Disjunction]
NON-OVERLAPPING EVENTS…
Two events that can not occur at the same time, the
probability that either A or B occurs is
P( A or B)  P( A)  P( B)
Concept
may be extended for more than 2 events.
P( A or B or C )  P( A)  P( B)  P(C )
6.5 Combining Probabilities
•Either/OR Probability… Example…
NON-OVERLAPPING EVENTS…
What is the probability of rolling a die and getting a 3, 4,
or 7?
1 1 0 2 1
P(3 or 4 or 7)     
6 6 6 6 3
6.5 Combining Probabilities
•Either/OR Probability:
OVERLAPPING EVENTS…
When two events are considered either/or, but
may occur at the same time, then the
probability that A or B occurs is:
P( A or B)  P( A)  P( B)  P( A and B)
6.5 Combining Probabilities
•Either/OR Probability:
OVERLAPPING EVENTS…
Consider this situation on tourism…Given the table, what is the probability
of meeting at random a person who is either a woman or French?
MEN
WOMEN
American
2
6
French
4
8
P( A or B)  P( A)  P( B)  P( A and B)
14 12 8 18
P( woman or French) 



 90%
20 20 20 20
6.5 Combining Probabilities
•Summary of Combining Probabilities:
AND
probability
Independent
events
AND
probability
Dependent
events
Either/OR
probability
Nonoverlapping
events
Either/OR
probability
Overlapping
events
P( A and B) 
P( A)  P( B)
P ( A and B ) 
P ( A)  P ( B given A)
P( A or B) 
P( A)  P( B)
P( A or B) 
P( A)  P( B)  P( A and B)
•QUESTIONS????
6.5 Combining Probabilities
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HOMEWORK:
pg 274 # 1-27 all