Independence (S.CPA.2 Wkst 1)

Transcription

HSS-CP.A.2 STUDENT NOTES WS #1
1
Independence
When working with probabilities we often perform more than one event in a sequence this is called a
compound probability. Compound probabilities are more complex than a single event probability to compute
because the first event might affect the probability of the second event happening.
For example, the probability of getting a head on a single flip of a coin is ½. If you flip the coin
and get a head, the second flip’s probability of getting a head is still ½ because the results of
the first flip does not in any way affect the second flip. The second flip has the exact same
probability as if it was the first flip, ½. When the first action does not affect the second
action’s probability in any way the events are known to be INDEPENDENT.
In contrast, if you have a jar of cookies with 7 chocolate chip cookies and 3 peanut
butter cookies the probability of getting a chocolate chip is 7/10 and the probability of
getting a peanut butter cookie is 3/10. Is the probability of getting a peanut butter
cookie still 3/10, if you first pick out a chocolate chip cookie and eat it? Of course not,
the probability of getting a peanut butter cookie now is 3/9 because a chocolate chip cookie is gone from the
jar. The second selection is affected by the first selection, thus these two events are NOT INDEPENDENT.
Definition: Two events, A and B, are independent if the fact that A occurs does not affect the probability of B
occurring (or vice versa).
Some other examples of independent events are:
-- Getting a head after tossing a coin AND selecting a purple marble from a bag.
-- Getting on head after tossing a coin AND rolling a 2 on a single 20-sided die.
-- Choosing a jack from a deck of cards, replacing it, AND then choosing a king as the second card.
What is the probability of rolling a 6 AND then getting a head on a coin flip?
These two events are independent - the die could roll any number of times and it would in no way influence
the flip of the coin. Let us use a set list, a tree and a Venn diagram to understand this problem.
Set U = {1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T}
Set S = Rolling a 6 = {6H, 6T}
Set H = Heads = {1H, 2H, 3H, 4H, 5H, 6H}
Set S ∩ Set H = {6H}
1
2
3
P(6 AND H) = 1 / 12
Remember that AND represents the intersection.
4
5
6
H
1H
T
1T
H
2H
T
H
T
2T
3H
3T
H
T
H
4H
4T
5H
T
5T
H
6H P(6H) =
T
6T
1
6
1
x
2
1
=
12
2
To find the probability of two independent events that occur in sequence, find the probability of each event
occurring separately, and then multiply the probabilities. This multiplication rule is defined symbolically below.
Note that multiplication is represented by AND.
When two events, A and B, are independent, the probability of both occurring is:
P(A and B) = P(A) · P(B)
Example #1
What is the probability of rolling a 6 and
then rolling a 5?
1
1
P(F) =
P(S) =
6
6
 1  1  1
P(S and F) = P(S) • P(F) =    =
 6  6  36
Example #3
What is the probability of getting a head on a coin
flip and then choosing a purple marble from a bag
that has 2 purple, 1 green, and 2 orange marbles?
These two events are independent. Let us use a tree
diagram to understand and solve this problem. What
is the P (Head and Purple)?
2
5
1
1
HEAD
2
5
2
PURPLE
1
P(HP) =
2
2
x
5
Example #2
Given a bag of marbles with 3 red, 2 green and 5
yellow. What is the probability of choosing a red,
replacing it, and then choosing a green?
3
2
P(R) =
P(G) =
10
10
6
 3  2 
P(R and G) = P(R) • P(G) =    =
 10  10  100
Example #4
A true and false question is followed multiple choice
question with possible four answers (1 correct & 3
wrong). What is the probability of getting both
questions correct, P(CC)?
2
=
1 1
1
1 COORECT
P(CC) = x =
2
4
8
MC
4
10
1
GREEN
2
CORRECT
T/F
ORANGE
5
2
5
1
1
TAIL
2
5
2
3 WRONG
MC
4
PURPLE
1 WRONG
T/F
2
GREEN
1 COORECT
MC
4
3 WRONG
MC
4
ORANGE
5
Understanding independence is critical to probability because we must always take into account how one
event affects the next event.
Determine if the following are independent or not.
a) Event #1
Event #2
b) Event #1
The Date
The Amount of Sunlight
Not Independent
Income
Event #2
Education Level
Not Independent
c) Event #1
Event #2
Your Height
Your Favorite Color
Independent
Determine the P (A and B) given that Event A and Event B are independent.
a) P(A) = 0.1, P(B) = 0.3
b) P(A) = 0..25, P(B) = 0.8
c) P(A) = 0.5, P(B) = 0.5
P(A and B) = 0.03
P(A and B) = 0.2
P(A and B) = 0.25
3
Mutually Exclusive and Independence
A common misunderstanding is that independence is the same thing as being mutually exclusive. I get why
this is confusing, to be independent in a typical English language context means to be alone or separate which
is basically what we understand mutually exclusive to mean. This definition of independence is NOT the
mathematical one. Independence is about whether one event affects another event’s probability or not.
Mutually exclusive sets are those that don’t share any elements and independent sets are those that don’t
impact each other’s probabilities.
Mutually exclusive is about the sharing of elements,
and independence, is about affecting each other.
In the example to the right, event A and event B are NOT MUTUALLY
EXCLUSIVE because there is an intersection between the two sets.
But Events A and Event B are INDEPENDENT because
P(A) • P(B) = (0.3)(0.5) = 0.15
P(A AND B) = 0.15
So here is one example where mutually exclusive and independence
obviously are two different things!!
Replacement and No Replacement
The terms replacement and no replacement get used a lot in compound probabilities problems because they
describe what you did with the first thing that you selected… did you put it back or did you keep it?
P (Getting a green marble, replacing it, and getting a green marble)
Independent
P (Picking a black queen, not replacing it, and getting an ace)
Not Independent
These two words are HUGE clues as to whether the events are going to be independent or not.
REPLACEMENT
Because the item is replaced, it resets the event back to the original arrangement and
no probabilities are altered. Thus REPLACEMENT tells us that the events are
INDEPENDENT.
NO REPLACEMENT
Because the item is NOT replaced, the probabilities are altered. Thus NO
REPLACEMENT tells us that the events are NOT INDEPENDENT.
a) picking a marble from a bag,
replacing the marble and then
picking again.
Independent
b) picking a marble from a bag and
then spinning a 4 color spinner.
Independent
c) picking a card from a standard
deck, not replacing the card and
then picking again.
Not Independent
4
Testing for Independence
We can use the formal relationship of P (A and B) = P(A) • P(B) to test independence. Here are three examples
of how we could use probabilities to determine if they are independent of each other.
Example #1
P(A) = 0.8
P(B) = 0.4
P(A and B) = 0.2
These are Not Independent because P(A) • P(B) = (0.8)(0.4) = 0.32 and this is not
the same as P(A and B) = 0.2 provided.
Example #2
P(A) = 0.6
P(B) = 0.5
P(A and B) = 0.3
These are Independent because P(A) • P(B) = (0.6)(0.5) = 0.30 and this is
the same as P(A and B) = 0.3 provided.
The same kind of test can occur but the information can be given in Venn diagram form.
Example #3
Check to see if P(A) • P(B) = P (A and B)
P(A) = 0.5
P(A) • P(B) = (0.5)(0.5) = 0.25
P(B) = 0.5
P (A and B) = 0.25
P(A AND B) = 0.25
INDEPENDENT
Example #4
P(A) = 0.7
P(A) • P(B) = (0.7)(0.2) = 0.14
P(B) = 0.2
P (A and B) = 0.14
P(A AND B) = 0.14
INDEPENDENT
Example #5
P(A) = 0.50
P(A) • P(B) = (0.5)(0.28) = 0.14
P(B) = 0.28
P (A and B) = 0.2
P(A AND B) = 0.20
NOT INDEPENDENT
a)
b)
c)
P(A) = 0.45
P(B) = 0.6
P (A and B) = 0.27
(0.3)(0.3) = 0.9
Not Independent
(0.25)(0.6) = 0.15
Independent
(0.45)(0.6) = 0.27
Independent
Name: _____________________________ Period ______
HSS-CP.A.2 WORKSHEET #1
1. Determine if the two events are independent of each other.
a)
Event #1
The Month
Event #2
The Temperature
Independent
Not Independent
b)
Your Height
Your Weight
Independent
Not Independent
c)
The day of the week Your hair color
Independent
Not Independent
d)
Your Age
Your height
Independent
Not Independent
e)
Your weight
Your income
Independent
Not Independent
a)
Event #1
Event #2
Choosing a marble from bag #1, and then choosing a marble from bag #2.
I or NI
b)
Selecting a marble from a bag, keeping it, and then selecting another marble.
I or NI
c)
Spinning a spinner to get a blue, and then flipping a coin to get a head.
I or NI
d)
Rolling an even number on a die, and then rolling it again to get a five.
I or NI
e)
Selecting a marble from a bag, replacing it, and then selecting another marble.
I or NI
3. The given two events, Event A and Event B are independent events.
a) P(A) = 0.4
P(B) = 0.3
P(A and B) = ________
b) P(A) = 0.76 P(B) = 0.11
P(A and B) = ________
c) P(A) = 0.2
P(B) = 0.2
P(A and B) = ________
d) P(A) = 0.55 P(B) = 0.1
P(A and B) = ________
a) P(A) = 0.4
P(A and B) = .22
P(B) = _______
b) P(A) = 0.74 P(A and B) = .37
P(B) = _______
c) P(A) = 0.85 P(A and B) = .51
P(B) = _______
d) P(A) = 0.9
P(B) = _______
P(A and B) = .45
5. Determine if the following are independent or not.
a) P(A) = 0.55 P(B) = 0.20
P(A and B) = 0.11
Independent
Not Independent
b) P(A) = 0.40 P(B) = 0.60
P(A and B) = 0.24
Independent
Not Independent
c) P(A) = 0.7
P(B) = 0.45
P(A and B) = 0.4
Independent
Not Independent
d) P(A) = 0.5
P(B) = 0.5
P(A and B) = 0.35
Independent
Not Independent
6. Travis says to a friend, I understand independence; it is when you have no elements in common. Is he
correct? Explain.
1
2
7. Determine if the following events are independent or not.
a) Independent or Not Independent
b) Independent or Not Independent
c) Independent or Not Independent
8. Determine if the event is independent or not, if the event is independent determine the probability of it
happening.
a) A bag of marbles has 3 red and 6 green marbles. What is the
probability of selecting two red with replacement?
Independent or Not Independent
If independent, P(R and R) = _______
b) A bag of marbles has 3 red, 1 green and 7 yellow marbles. What Independent or Not Independent
is the probability of selecting a green and then a yellow with
replacement?
If independent, P(G and Y) = _______
c) There are two bags of marbles, in Bag #1, there are 3 red and 2
green, and in Bag #2, there are 2 red and 6 green. What is the
probability of selecting a green from Bag #1, and a red from Bag
#2?
If independent, P(G and R) = _______
d) A bag of marbles has 1 red, 1 green and 3 yellow marbles. What Independent or Not Independent
is the probability of selecting a yellow and then a yellow without
replacement?
If independent, P(Y and Y) = _______
e) Given a standard deck of cards. What is the probability of
selecting a jack and then an ace without replacement?
If independent, P(J and A) = _______
f) A spinner has four equal color (Red, Green, Yellow, Blue)
quadrants and a die has 12 sides. What is the probability of
getting blue on the spinner and a factor of 12 on the die?
g) You roll one sixed sided dice twice. What is the probability of
getting a six and then a value less than 3?
If independent, P(B and F) = _______
If independent, P(S and L) = _______
9. How does the term replacement help keep events independent of each other?
3
10. Why can’t P(A and B) ever be greater than P(A)?
11. Events A and Event B are independent. Complete the Venn diagram and determine the probability.
a) P(A and B) = 0.3
b) P(A and B) = 0.22
P(A and Not B) = 0.2
P(B and Not A) =
0.18
P(A) = ____________
P(B) = ___________
P(B) = ____________
P(A) = ___________
c) P(A and B) = 0.6
P(A) = ____________
P(A and Not B) = 0.2
P(B) = ____________
P(B and Not A)? __________
d) P(A and B) = 0.4
P(A) = 0.5
P(B) = ____________
P(B and Not A)? __________
P(Not A and Not B)? ______
e) P(A) = 0.32
P(A and B)? ______
P(B) = 0.25
P(B and Not A)? __________
P(A and Not B)? __________
f) P(A and B) = 0.2
P(A)? ______
P(B) = 0.8
P(B and Not A)? __________
P(A and Not B)? __________
f'
H SS-CP.A.2 WO RKS H E ET #7
Period
1. Determine if the two events are independent of each
a)
b)
c)
d)
e)
Event #1
Event #2
The Month
The Temperature
lndependent
Your'Height
Your Weight
Independent
Not lndependent
nd
Not lndependent
Your height
lndependent
Not lndependent
Your income
lndependent
The day of the
week Your hair color
Age
Your weight
Your
ffi
@
\---j-_-_-
Event #1
a)
b)
c)
d)
e)
3. The given
Event #2
choosing a marble from bag #1, and then choosing a marble from bag#Z.
lorNl
Selecting a marble from a bag, keeping it, and then selecting another marble.
ror@
Spinning a spinner to get a blue, and then flipping a coin to get a head.
lorNl
eor
Rolling an even number on a die, and then rolling it again to get a five.
Selecting a marble from a bag, replacing it, and then selecting another marble.
two events, Event A and Event
B are
lorNl
independent events.
a) P(A) =
6.4 P(B) = g.3
P(A and B)
= 0 '11-
b) p(A) = 016 p(B) = 9.11
P(A and B) =
c) P(A) =
g.2
P(A and B)
= O,0u?
d) p(A) = 0.55 p(B) = 6.1
P(A and B) =
P(B) =
9.2
Nl
a) P(A) =
O.+ P(A and B\ = .22
c) P(A)= 0.85 P(A and B) =
.51.
P(B)
= 0,5,
P{B)=
CI,{"
b) P(A) = 014 p(A and B) = .37
P(B) =
d) P(A) = 9.9 P(A and B) = .45
P(B) =
5. Determine if the following are independent or not.
6.26
0.tt
ndependenf
Not lndependent
b) P(A) = 0.40 P(B) = 9.69 P(A and B) = 0.24
lndependent
Not lndependent
P(A and B) = 0.+
lndependent
ot lndependent
P(A and B) = 0.35
lndependent
Not lndependent
a) P(A) =
c) P(A) =
0.55
6.7
P(B) =
P(B) =
9.45
d) P(A) = 9.5 P(B) = 9.5
P(A and B) =
6. Travis says to a friend, I understand independence;
correct? Explain.
it
is when you have no elements in
common. ls he
l'
HSS.CP.A.z WO RKSH E ET #7
7. Determine if the following events are independent or not.
a)flndependenlpr Not lndependent
---
b)lndependentorNotlndependent.@orNotlndependent
g. Determine if the event is independent or not, if the event is independent determine the probability of it
happening.
a) A bag of marbles has 3 red and 6 green marbles.
what
lndependent or Not lndePendent
is the
probability of selecting two red with replacement?
lf independent, P(R and R) =
b)Abagofmarbleshas3red,1greenand7yellowmarbles.What@orNotlndependent
is
the probability of selecting
a green and
reptacement?
then a yellow
with
lf independent, P(G and Y)
GXfi
two bags of marbles, in Bag #1, there are 3 red and
green, and in Bag#2,there are 2 red and 6 green. What is the
c) There are
2
=
a /,_ ,
t t Lt
lndependent or Not lndependent
probabilityofselectinga8reenfromBag#1.,andaredfromBaglfindependent,P(GandR)
#2?
d)Abagofmarbleshas1red,1greenand3yellowmarbles.Whatlndependent",@
is
withqgt
a yellow
LrrErr q
and then
yErrrJw orru
a yellow
selecting d
vLrrvuuiijthe probability of seleLtrrrE
repracement?
e) Given a standard deck of cards. What is the probability
selecting a jack and then an ace without replacement?
of
f) A spinner has four equal color (Red, Green, Yellow, Blue)
quadrants and a die has 12 sides. What is the probability of
getting blue on the spinner and a factor of 12 on the die?
f ,r,)
t
,q ,
b, tz\
lf independent, p(y and y) = i/ fr
lndependent or Not lndependent
lf indePendent,
@or
P(J and A) =
Not lndependent
tf independent, P(B and F)
- blqf
(r)(a
g) you roll one sixed sided dice twice. what is
getting a six and then a value less than 3?
the probability of
lndependent or Not lndePendent
lf independent, P(S and L) =
9. How does the term replacement help keep events independent of each other?
Rop
\o**-,
-
-\ rt54k e-rr'^/Stk' v*->
i'a{'qr*-'d--+
E
['
H SS-CP.A.z W O RKSH E ET #7
3
10. Why can't P(A and B) ever be greater than P(A)?
11. Events A and Event B are independent. Complete the Venn diagram and determine the probability.
a) P(A and B) = 9.3
b) P(A and B) = O.22
P(A and Not B) = Q.2
P(B and Not A) =
P(A)= 0'5
0.18
P(B)=
P(B)=
D.{a
'::i:::r'.',!*rY
9.6
P(A and Not B) = Q.2
c) P(A and B) =
=
P(B) =
P(A)
,h)=# =oL
O
o.-rf
P(B and Not
d) P(A and B) = 9.4
P(A) = 9.5
,7
A)? b.tr
P(A) =
P&),?ln)=P(*ao\
e.s.Pb) - o'6
rb) = ff= o'1f
o3f 'o,b= a'tf
P(B) =
P(B and Not A)?
P(Not A and Not B)?
e) P(A) = 0.32
P(A and Bl?
0,09
P(B) = 6.25
P(B and Not
A)? O,l7
P(A and Not
B)? O'L'l
_
f) P(A and B) = 9.2
P(A)?
P(B) = g.g
P(B and Not A)?
P(A and Not B)?
?(*)'Pb)= P(Gqv)
DiL -(o.'r)= n(*as)
o,at
: p 1*tf)

Independence (S.CPA.2 Wkst 1)

Transcription

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