Download Independent/Dependent/Conditional Events

Aim: How do we determine the
probability of compound events?
Do Now:
What is the probability of flipping a regular
coin and getting a head? P(H) = 1/2
Inside a baggie are 1 red, 1 pink, 1 blue and
1 green chip. What is the probability of
reaching into the baggie and picking a
green chip? P(G) = 1/4
What is the probability of reaching into the
baggie and picking a green chip and
flipping a coin and getting heads?
P(G
and H) =
1/2• 1/4 = 1/8
Describe how the first two problems are
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
different from the third.
Construct a tree diagram to show all
possible outcomes for the 3rd part of the
Do Now problem. Find P(pink, H)
Event A
Pick Chip
pink
red
Event B
8 outcomes
Flip Coin Sample Space
pink H
H
pink T
T
1
P(p H) =
8
H
red H
T
red T
blue
H
T
blue H
blue T
green
H
T
green H
green T
P(pink)
P(H) =
P(pink, H) =
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
= 1/4
1/2
1/4 • 1/2 = 1/8
I
n
d
e
p
e
n
d
e
n
t
Compound Event
Compound Event - two or more activities
Ex. Rolling a pair of dice
What is the probability of rolling a pair of
dice and getting a total of four?
6
5
4
Die 1 3
2
1
7
6
5
4
3
2
8
7
6
5
4
3
1
2
9 10 11 12
8 9 10 11
7 8 9 10
6 7 8 9 P(4) = 3/36
5 6 7 8
= 1/12
4 5 6 7
3 4
Die 2
5
6
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
Independent Events
Independent Events – two events are
independent if the occurrence of either of
them has not effect on the probability of the
other.
Mutually exclusive – two events A & B are
mutually exclusive if they can not occur at
the same time. That is, A and B are
mutually exclusive when A  B = 
Independent events can occur at the same or
different times, and have no effect on each other.
Ex. mutually exclusive?
rolling a 2 and a 3 on a die
Yes
rolling
an even number or a multiple
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
of 3 on a die
No
Probability of Two Independent Events
The probability of two independent events can be
found by multiplying the probability of the first
event by the probability of the second event.
AND Probabilities with Independent Events
If A and B are
events,
then
P(Aindependent
and B) = P(A)
· P(B)
P(A and B) = P(A) · P(B)
Ex: A die is tossed and a
spinner is spun.
What’s the probability of
throwing a 5 and spinning
red? P(5 and R)?
Faster than
drawing a
tree
diagram!!
Event A
Event B
1

P (5)
6
1
P ( red ) 
4
1
1
1
P(5 and Red) =


6
4
24
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
Model Problem
A special family has had nine girls in a row. Find
the probability of this occurrence.
Having a girl is an independent event
with P(1 girl) = 1/2
Probability of two Independent Events
P(A and B) = P(A) · P(B)
extends to multiple independent events
1 1 1 1 1 1 1 1 1
P (nine girls in a row) =
2 2 2 2 2 2 2 2 2
9
1
1
  
512
 2
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
Model Problem
If the probability that South Florida will be hit by a
hurricane in any single year is 5/19
a) What is the probability that S. Florida will be hit
by a hurricane in three consecutive years?
3
5 5 of
5 event
125the
 5 (A)
 plus
•
The
probability
P (hurricane - 3) =
  
 0.018
probability of19"not
1:
19 A”
19 or ~A,
19  equals
6859
P(A) + P(~A) = 1; P(A) = 1 – P(~A);
b) What is the probability that S. Florida will not be
P(~A) = 1 – P(A)
hit by a hurricane in the next ten years?
P (no hurricane) = 1  P (hurricane)
5 14
 1

 0.737
19 19
10
10
 14 
P (no hurricane 10 yrs) =     0.737   0.047
Aim: ‘And’ Probabilities &
19  Events Course: Math Lit.
Independent
Not So Independent!
There are 4 red, 3 pink, 2 green and 1 blue
chips in a bag. What is P(pink)? 3/10
What is the probability of picking a pink and
then reaching in and picking a second pink w/o
replacing the first one picked?
3
2
nd
P ( pink ) 
P (2 pink w/o replacement ) 
10
9
6/90 or 1/15 is the probability of picking a pink
chip and then picking a second pink chip.
BUT ONLY IF THE FIRST PINK CHIP
WAS NOT RETURNED TO THE BAG.
The selection of the second event was affected
by the selection of the first.
Dependent
Events
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
Dependent Events
Two events are dependent events if the
occurrence of one of them has an effect on
the probability of the other.
AND Probabilities with Dependent Events
If A and B are dependent events, then
P(A and B) = P(A) · P(B given that A has
occurred)
extends to multiple dependent events
You are dealt three cards from a 52-card deck.
Find the probability of getting 3 hearts.
P(1st heart) = 13/52
P(2nd heart) = 12/51
P(3rd heart) = 11/50
Probabilities
& Independent
Events Course:
Math Lit.
P(hearts)Aim:
= ‘And’
13/52
· 12/51
· 11/50
= 1716/162600
 0.0129
Model Problem
Three people are randomly selected, one
person at a time, from 5 freshman, two
sophomores, and four juniors. Find the
probability that the first two people selected
are freshmen and the third is a junior.
P(1st selection is freshman)
= 5/11
P(2nd selection is freshman) = 4/10
P(3rd selection is junior)
= 4/9
P(F, F, J) = 5/11 · 4/10 · 4/9 = 8/99
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
Model Problem
Find the probability of
choosing two pink chips
without replacement.
Event A
Event B
Dependent Events
P(pink)
P(pink)
P(pink, pink) =
3/10
2/9
3/10 • 2/9 = 6/90
or 1/15
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
Counting Principle w/Probabilities
Model Problem
Find the probability of
choosing blue and then a
red chip without
replacement.
Event A
Event B
P(blue)
P(red)
1/10
4/9
Dependent Events
P(blue, red) =
1/10 • 4/9 = 4/90
or 2/45
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
Counting Principle w/Probabilities
Probability of Dependent Events
1. Calculate the probability of the first event.
2. Calculate the probability of the second
event, etc. ... but NOTE:
The sample space for the probability of the
subsequent event is reduced because of the previous
events.
3. Multiply the the probabilities.
Ex. A bag contains 3 marbles, 2 black and one white.
Select one marble and then, without replacing it in the bag,
select a second marble. What is the probability of
selecting first a black and then a white marble?
Event A
Event B
2
3
1
P(A) =
P(B) = 2
2 1 2 1
P(Black 1st, White 2nd =   
3 2 6 3
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
Key words - without replacement
Model Problem
From a deck of 10 cards (5 ten-point
cards, 3 twenty-point cards, and 2
fifty-point cards), Ronnie can only
pick 2 cards. In order to win the
game, he must pick the 2 fifty-point
cards. What is the probability that he
will win?
10
10
10
10
10
20
20
20
10
20
10
50
20
10
50
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
10
50
Model Problem
From a deck of 10 cards Ronnie can
only pick 2 cards. In order to win the
game, he must pick the 2 fifty-point
cards. What is the probability that he
will win?
10
10
10
10
10
20
20
20
10
20
10
50
20
10
50
10
50
Dependent
Event A
Event B
P(50, 50) =
P(50) =
P(50) =
2/10 • 1/9 =
2/90 = 1/45
2/10
1/9
Counting Principle
Aim: ‘And’ Probabilities & Independent w/Probabilities
Events Course: Math Lit.
Model Problems
Penny has 3 boxes, each containing 10 colored balls.
The first box contains 1 red ball and 9 white balls,
the second box contains 3 red balls and 7 white balls,
and the third box contains 7 red balls and 3 white
balls. Penny pulls 1 ball out of each box.
Box 1
Box 2
Box 3
A. What is the probability that Penny pulled
3 red balls?
P(r,r,r) = 1/10 • 3/10 • 7/10 = 21/1000
B. If Penny pulled 3 white balls and did not
replace them, what is the probability that
she will Aim:
now
pull 3 red balls?
‘And’ Probabilities & Independent Events Course: Math Lit.
P(r,r,r) = 1/9 • 3/9 • 7/9 = 21/729
Model Problems
A sack contains red marbles and green marbles. If
one marble is drawn at random, the probability that
it is red is 3/4. Five red marbles are removed from
the sack. Now, if one marble is drawn, the
probability that it is red is 2/3. How many red and
how many green marbles were in the sack at the
start?
x = original red marbles 5
y = original number of green marbles 15
3
x__
2
x-5
=
=
4
x+y
3
x+y-5
3x + 3y = 4x
3y = x
2x + 2y - 10 = 3x - 15
2y + 5 = x
3y = 2y + 5
y= 5
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
3y = x = 15
Conditional Probability of A and B
Conditional Probability - The probability of
event B, assuming that the event A has
already occurred, is call the conditional
probability of B, given A. This is denoted by
P(B|A).
number of outcomes common to B and A
P ( B | A) 
number of outcomes in A
n( A  B )

n( A)
# outcomes of B that are in the
restricted sample space of A
P(B|A) =
# outcomes in the restricted
sample space of A
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
Model Problem
Find the probability of rolling a die and getting a
number that is both odd and greater than 2.
n(odd ) 3
P (odd ) 

n( S )
6
{1, 3, 5}
n( 2) 4
P ( 2) 

n( S )
6
{3, 4, 5, 6}
{3, 5}
2
2
P (odd   2) 

n( S ) 6
Conditional probabilities are calculated based on
common outcomes regarding the two events A
and B; A  B.
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
Conditional Probability of A and B
recall:
The intersection of sets A and B, denoted by
A  B, is the set consisting of all elements
common to A and B.

A  B = {x|x  A AND x  B}

U
Region
II
A
Reg.
III
Region
IV
B
Region I
The intersection of sets A and B is region III.
‘and’ is the term used to describe intersection
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
Venn Diagrams
Find the probability of rolling a die and getting a
number that is both odd and greater than 2.
>2
odd
1
3
5
3
5
P(odd) = 3/6
2
1
4
6
P(> 2) = 4/6
3
5
4
6
P(odd  > 2)
= 2/6
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
Model Problem
A letter is randomly selected from the letters of the
English alphabet. Find the probability of selecting
a vowel, given that the outcome is a letter that
precedes h.
number of outcomes common to B and A
P ( B | A) 
number of outcomes in A
n( A  B )

n( A)
sample space
S = {a, b, c, d, e, f, g}
2
P (vowel | letter precedes h) 
7
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
Model Problem
The table below shows the differences in
political ideology per 100 males and per
100 females in the 2000 US presidential
election. Find the probability thatn(the
A  B)
P ( B | A) 
person
n( A)
a. is liberal, given that the person is female
20 1

P ( liberal | female ) 
100 5
b. is male, given that the person is
conservative.
39
P ( male | conservative ) 
64
Liberal Moderate Conservative
male
female
16
45
39
20
55
25
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.
= 100
Model Problems
A pair of dice are tossed. Find the
probability that the sum on the two dice is 8,
given that the sum is even.
A pair of dice are tossed twice. Find the
probability that both rolls give a sum of 8.
Liberal Moderate Conservative
HS only
7
35
13
College
10
15
20
Find the probability that 1 person selected is
moderate,
given college attendance.
Aim: ‘And’ Probabilities & Independent Events Course: Math Lit.