Transcript Probability

N!

A!*B!*C!
Quiz 10-3
1. You have the following bills in your wallet: three $20’s,
four $10’s, five $5’s, and six $1’s.
What is the number of distinct ways you could pay out the
bills one at a time?
18!
 514,594,080

3!*4!*5!*6!
2. How many distinct license plates can be made using
2 digits (numerals 0 – 9) and 4 letters ( a – z) ?
##LLLL
 10 *10 * 26 * 26 * 26 * 26  45,697,600
3.
What is the probability of getting 4 aces in a randomly dealt hand of 4
cards?
C
1

4
4
52
C4

270,725
 0.0000037
10-4
Disjoint and Overlapping Events
Theoretical Probability
The probability of an event occurring:
# of ways to achieve that event
P(event) 
total # of possible outcomes
There are 4 different colored marbles in a bag (red, blue, green
and clear). What is the probability of pulling out a red one
on the first try?
# of red marbles in the bag
P(red ) 
total # of marbles in the bag
1
P (red )   0.25
4
Vocabulary
5 red,
4 blue, 3
compound events: more than one
event occurring.
picking 2 red OR 1 green.
green
picking 2 red AND 1 green.
Bag of marbles
Is there difference between OR and AND ?
Which has the higher probability, OR and AND ?
“Disjoint events: (mutually exclusive). The chance of one
event does not affect the chance of another event.
2 red,
4 red,
3 green
3 green
Bag #1
Bag #2
Event #1: picking a red marble from the first bag and
Event #2 picking a green marble from the second bag.
“Disjoint events: (mutually exclusive). The chance of one
event does not affect the chance of another event.
2 red,
3 green
Bag of marbles
Event #1: picking a red marble from the bag
2
P (red ) 
5
Event #2 replacing the red marble then picking a green
marble from the bag.
3
P ( green ) 
5
“Overlapping events: (not mutually exclusive). The chance
of one event is affected by the preceding event.
2 red,
3 green
Bag of marbles
Event #1: picking a red marble from the bag
2
P (red ) 
5
Event #2 without replacing the red marble, picking a
green marble from the bag.
3
P ( green ) 
4
Venn Diagrams
Disjoint events: they
have nothing in common.
Group A
Elements of group A (only):
Number of elements in group A: 4
Elements of group B (only):
Number of elements in group B: 2
Total Number of elements in groups A and B: 6
Group B
Probability of disjoint events.
“disjoint” (events don’t overlap)
P(A or B) = P(A) + P(B)
3
Black
Hair
4
Red
Hair
3 4
P (black or red hair)    1
7 7
Your turn:
1. There are 20 people in an English class. Five are
German, 12 are Mexican, and 3 are Italians. What is the
probability of being either Mexican or Italian if you are in
this class?
12 3
3
15

20

20

20

4
 0.75
2. What is the probability of dealing either a King or a
Queen from a deck of cards if you deal one card only?
C1
4 C1


52 C1
52 C1
4
4
4
2


  0.154
52 52 13
Venn Diagrams
Group A
Overlapping events: they
have something in common.
Group B
Elements in group A
Elements in A: 4
Elements in group B
Elements in B: 4
Elements in group A or B
Elements in both
A and B: 6
Why can’t you add the number in A and the number in B
to get the total number of elements?
Double-counting the overlap!
Total number = (# in A) + (# in B) – (# in overlap)
Probability of overlapping events.
Total number = (# in A) + (# in B) – (# in overlap)
P(A or B) = P(A) + P(B) – P(A and B)
Blonde
Hair
Bill
Jim
Amber
Maria
Angelica
Girls
P(blond or girl)  ?
P(A or B)= P(A) + P(B) – P(overlap)
P(blond or Girl) = P(blond) + P(girl) – P(blond and girl)
3 3 1
5
P (blond or girl)   

5 5 5
5
Probability of overlapping events.
# hearts = ? = 13
# face cards = ? = 12
total number of cards = ? = 22
# hearts and face cards = ? = 3
Face
cards
Hearts
P(heart)  ?

13
22
P(face card)  ?
A,
2,3,4,5,6,
7,8,9,10
Hearts:
J, Q, K
Spades: J, Q, K
Clubs: J, Q, K
Diamonds: J,Q,K
3
12
P
(
heart
and
face
card)

?


22
22
P(A or B) = P(A) + P(B) – P(A and B)
Probability > 1
13 12 25
P (H or FC ) 


 the universe just “imploded”
22 22 22
22
13 12 3

P(H or FC ) 


22
22 22 22
Probability of overlapping events.
13
P(heart) 
22
Hearts
A,
2,3,4,5,6,
7,8,9,10
3
12
P(heart and face card) 
P(face card) 
22
22
Face
cards
What does “OR” mean?
Hearts: Spades: J, Q, K
J, Q, K Clubs: J, Q, K
Diamonds: J,Q,K
“OR”: one of the two conditions is met, NOT BOTH conditions.
We need to subtract out the probability of both conditions occurring.
P(H or FC)  P(H)  P(FC) - P(H and FC)
13 12 3
P (H or FC)  1
P(H or FC) 

22 22 22
Probability of disjoint or overlapping events.
“disjoint” (mutually exclusive):
P( A or B)  P(A)  P( B)
“overlapping”
P( A or B)  P(A)  P( B)  P( A and B)
Using the formulas:
P( A or B)  P(A)  P( B)
P( A or B)  P(A)  P( B)  P( A and B)
“disjoint” (mutually exclusive):
“overlapping”
P(A) = 0.5
P(A or B) = ?
P(A) = 0.2
P(A and B) = ?
P(B) = 0.35
P(A and B) = 0.2
P( A or B)  P(A)  P( B)  P( A and B)
P( A or B)  0.5  0.35  0.2  0.65
P(B) = 0.6
P(A or B) = 0.7
P( A or B)  P(A)  P( B)  P( A and B)
Solve for P(A and B)
0.7  0.2  0.6  P( A and B)
P( A and B)  0.2  0.6  0.7
P( A and B)  0.1
Your turn:
“overlapping”
3.
P( A or B)  P(A)  P( B)  P( A and B)
P(A) = 0.8
P(A or B) = ?
P(B) = 0.25
P(A and B) = 0.1
P( A or B)  P(A)  P( B)  P( A and B)
P( A or B)  0.8  0.25  0.1
4.
P(A) = 0.37
P(A and B) = ?
P(B) = 0.49
 0.95
P(A or B) = 0.65
P( A or B)  P(A)  P( B)  P( A and B)
Solve for P(A and B) 0.65  0.37  0.49  P( A and B)
P( A and B)  0.37  0.49  0.65
P( A and B)  0.21
Theoretical Probability
The probability of an event occurring:
# of ways to achieve that event
P(event) 
total # of possible outcomes
The challenge you have is counting the ways that define success
and then counting the total possible outcomes.
What is the probility of pulling an A, followed by a B, and then
a C out of a bag with the letters ‘A’, ‘B’, and ‘C’ in it ?
# of ways to draw A, B, then C
P( A, B, C ) 
total # of ways to draw the 3 letters out of the bag
3 choose 3
P( A, B, C ) 
3 permutate 3
1

3!
1
  0.1 6
6
Your Turn:
# of ways to achieve that event
P(event) 
total # of possible outcomes
5. What is the probality of being dealt a 5 card hand that
has 3 kings and 2 Aces?
6. What is the probability of picking 1 red or a green marble
from a bag containing 2 red and 3 green marbles?
7. In a room of people, there are 15 boys and 17 girls. 5 of
the boys have blond hair and 4 or the girls have blond
hair. What is the probability of having blond hair?
8. In a room of people, there are 20 boys and 18 girls. 8 of
the boys have black hair and 4 or the girls have black
hair. What is the probability of not having black hair and
being a boy?
Geometric Probability
9.
What is the probability of hitting the pink ring?
10.
11.
What is the probability of hitting either the pink or dark blue ring?
What is the probability of not hitting the pink ring?
125
75
25
Your Turn:
# of ways to achieve that event
P(event) 
total # of possible outcomes
12.
At the Roy High School Talent show 7 musicians are scheduled to
perform. They are: Bill, Brad, Bob, and Brody (boys) and Kylee, Kaylee,
and Kyla (3 girls). What is the probability that 2 boys will be first?
P(2 boys 1st ) 
# ways to arrange 2 of 4 boys in order
# of ways to arrange 2 (of 7) people in order
4 choose 2
P(2 boys 1 ) 
7 choose 2
st
C2

7 C2
4
6

21
 0.29
Your Turn:
# of ways to achieve that event
P(event) 
total # of possible outcomes
13.
What is the probability of getting 3 aces in a randomly dealt hand of 7
cards?
# ways to get 3 aces out 4 cards
P(4 aces) 
# of ways to get 7 cards out of 52.
4 choose 3
P(r aces) 
52 choose 7
C3
4


133,784,560
52 C7
4
 0.000 000 03
Your Turn:
# of ways to achieve that event
P(event) 
total # of possible outcomes
14. The lottery uses numbers 1 thru 46.
6 numbers are drawn randomly.
The order in which you choose the numbers doesn’t matter. What is
the probability of winning the lottery if you buy one ticket (assume
nobody else picks the winning number) ?
P (6 # ' s ) 
# ways to get the 6 correct numbers
# of ways to pick 6 of 46 numbers
How many ways can you get the 6 out of 6 correct numbers?
How many ways can you pick 6 of 46 numbers?
Is picking 6 of 46 a permutation or a combination?
1
6 choose 6
P( win ) 

46 choose 6 46 C6
1

9,366,819
 0.00000011
The probability of something NOT happening.
The probability that it will rain is 50%.
What is the probability that it won’t rain?
The probability that it will rain is 20%.
What is the probability that it won’t rain?
P( A )
P (rain )  50%
P(r a i n )  100%  50%
P(rain )  20%
P(r a i n )  100%  20%
Means: “the probability that event A will not happen.
Probability for the Sum of Two Fair Dice
#..ways..to..get..a..7
P ( e) 
total..#..of .. possible ..rolls
Die
Roll
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Sums
P (e) 
Sums
6 1
  0.1 6
36 6
Using Venn Diagrams:
In a parking lot, 60% of the cars are Fords and the rest are of other makes. 30% of
the Fords are white. 50% of all the cars are white.
% of cars that are white fords = 0.3 * 0.6 = 0.18
What percentage of the cars that are not Fords are not white? 8%
If a car were chosen randomly, what is the probability that it is not
a ford and is white?
32%
Other non-Ford,non-white cars 1.0 - 0.42 - 0.18 – 0.32 = 0.08
Fords: 0.6
Fords (not
white)
0.6*0.7 =0.42
White
Fords
0.18
White
Cars: 0.5
White nonfords:
0.5 – 0.18
=0.32