section 5.1.a basic concepts of probability today, you will learn to… > identify the sample...
TRANSCRIPT
Section 5.1.A
Basic Concepts of Probability
Today, you will learn to…
> identify the sample space of a probability experiment
Sample Space
all possible outcomes of a probability experiment
Roll a die: { 1, 2, 3, 4, 5, 6 }Flip one coin: { T, H }
Gender of one child: { M, F }
Event one possible outcome of a
trial in an experiment
Roll an even number { 2, 4, 6 }
Get tails when flipping a coin { H }
Draw a queen from a deck of cards{Q, Q, Q, Q}
If P(E) = 0 the event is impossible
the event is certainIf P(E) = 1
0 < P(E) < 1 or
0% < P(E) < 100%
Can probability be…
zero?
- 2%?
2.7 ?
¾ ?
YES
NO, negative
NO, greater than 1
YES
A probability experiment consists of tossing a coin and then rolling a six-sided die. Identify the sample
space.
H T
1 2 3 4 5 6 1 2 3 4 5 6
The sample space has 12 outcomes.
{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
Draw a tree diagram showing the sample space of the gender sequence of a family with 3
children.B G
B G B G
B G B G B G B G
{BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}
Draw a tree diagram showing the sample space of 4 days of weather (sunny or rainy).
S R
S R S R
S R S R S R S R
S R S R S R S R S R S R S RS R
S R
S R S R
S R S R S R S R
S R S R S R S R S R S R S RS R
List the sample space of 4 days of weather (sunny or rainy).
SSRS, SSSS, SSSR, SSRR,
S R
S R S R
S R S R S R S R
S R S R S R S R S R S R S RS R
{SSSS, SSSR, SSRS, SSRR, SRSS, SRSR, SRRS, SRRR,RSSS,RSSR, RSRS, RSRR,RRSS, RRSR, RRRS, RRRR}
16 different possibilities
Simple Event consists of one single
outcome
Simple: 65 in tall
Not Simple: over 6’ tall
Decide whether the event is simple or not.
the student's age is between 18 and 23
the student’s age is 20
not simple
simple
Decide whether the event is simple or not.
the student scored an 85%
the student scored a B
not simple
simple
Let’s Practice!!!!
Lesson 5.1.BTypes of Probability
Theoretical probability
Subjective probabilityExperimental probability
Theoretical probabilityis used when we already have the data we need to
find the probability.
dice, raffle, cards, coins, etc.
P (selecting a 7 of diamonds) =
P ( selecting a diamond) =
You select a card from a standard deck. Find the probability of the
following.
152
13 52
=
=
0.019 =
2%
25%
0.25 =
12 16
6 16
P(R or G) =
P (not R) =
A bag contains 16 marbles: 10 blue (B), 4 red (R), and 2 green (G). One marble
is randomly drawn from the bag.
0.75
=
=
=0.375
=
38%
75%
Experimental Probabilityis based on data collected
in an observation or experiment .
You actually do an experiment to find the probability
An insurance company analyst determines that in every 200 claims,
4 are fraudulent. What is the probability that the next claim the company processes is fraudulent?
4 200 = 0.02 = 2%P(fraudulent)=
A pond contains 3 types of fish.You catch 40 fish and record the
type. You catch 13 bluegill, 17 redgill, and 10 catfish. If you
throw all of the fish back and catch another fish, what is the probability that it is a catfish?
10 40
= 0.25 = 25%P(catfish)=
Toss a coin 10 times and count the number of times you get heads.
Find the class experimental probability.
Coin Tossing Probability Experiment
P(getting heads) = 105
= 50%
theoretical probability
If you repeat a probability experiment over and over, the
experimetnal probability of an event will equal the theoretical probability
of the event.
Law of Large Numbers
P(15 - 24 years old) =
Employee data was collected Employee Age Frequency, f
15 – 24 54
25 - 34 366
35 - 44 233
45 - 54 180
55 - 64 125
65 and over 42
541000
=0.054
=5%
Subjective probability
Results from intuition, educated guesses, and
estimates.
Subjective probabilityA doctor may feel that a patient has
a 90% chance of a full recovery.
A business analyst may predict that there is a 0.25 chance of
decreased sales.
A weather reporter makes an educated guess that there is a
20% chance of rain today.
Theoretical, Experimental, or Subjective?
The probability of your phone ringing during class is 0.15
subjective probability because it is most likely based on an
educated guess
Calculate the probability of the event & its complement.
E: Pick a red card
E: Roll a 3 or greater
P(red) =
P(black) =
50%
P(>3) = =67% P(<3) = 33%
50%E’:Pick a black
E’: Roll a 1 or 2
46
The probability that a voter chosen at random will vote
republican is 45%
statistical probability because it is most likely based on a survey of a
sample of voters
Theoretical, Experimental, or Subjective?
theoretical probability because you know the number of outcomes and
each is equally likely
The probability of winning a 1000-ticket raffle with one
ticket is 1 in 1000.
Theoretical, Experimental, or Subjective?
The complement of event E (E’) is the set of all outcomes in a
sample space that are
NOT included in event E.
P(E) + P(E’) = 1
P(E) + P(E’) = 100%
Identify the complement of the event. Give both probabilities.
E: Roll a 4
E: Roll an odd number
E’:
E’:
P(E) = P(E’) =
P(E) = P(E’) =
16
56
3612
12
roll a 1,2,3,5,or 6
roll an even
Practice Time!!!!
Lesson 5.2.A
Fundamental Counting
PrincipleToday, we will learn to… > use the Fundamental Counting
Principle
How many ways I can put together outfits with 3 pairs of
pants (jeans, black, and tan) and 4 shirts (white, purple, red,
and teal)?
Jeans
Black
Tan
12 outfits
whitepurpleredteal
whitepurpleredteal
whitepurpleredteal
Find the number of ways I can put together a sundae with 3 kinds of ice cream (van, choc, swirl), 3 different
toppings (caramel, strawberry, butterscotch) and 3 different crunchy toppings (choc chips, M&Ms, oreo).
Van Choc Swirlcaram
el
strawb
erry
bu
tterscotch
caramel
strawb
erry
bu
tterscotch
caramel
strawb
erry
bu
tterscotch
Choc. C
hipM
&M
Oreo
Choc. C
hipM
&M
Oreo
Choc. C
hipM
&M
Oreo
Choc. C
hipM
&M
Oreo
Choc. C
hipM
&M
Oreo
Choc. C
hipM
&M
Oreo
Choc. C
hipM
&M
Oreo
Choc. C
hipM
&M
Oreo
Choc. C
hipM
&M
Oreo
27 different sundaes
The Fundamental Counting Principle
If one event can occur in A ways, another event can occur in B ways, another event can occur in C ways,and so on, then the total number of
possible outcomes isA∙B∙C∙…
The rule can be extended for any number of events.
You are purchasing a new car. You can choose from 4 different manufacturers,
3 different car sizes, and 6 different colors. How many different ways can you select one manufacturer, one car
size, and one color?
72 cars_·_·_ =4 3 6
The access code for a car’s security system consist of four digits. Each digit can be 0 through 9. How many access codes are possible if each digit can be
used only once and not repeated?
__·__·__·__ = 504010 9 8 7
The access code for a car’s security system consist of four digits. Each digit can be 0 through 9. How many access codes are possible if each digit can be
repeated?
__·__·__·__ =10 10 1010 10,000
How many license plates can you make if a license plate consists of six
letters that cannot be repeated?
__·__·__·__·__·__=26 165,765,60025 24 23 22 21
How many license plates can you make if a license plate consists of six letters that can be repeated?
__·__·__·__·__·__=26 308,915,77626 26 26 26 26
Let’s Practice!!!!
Lesson 5.2.B
Permutations
Today, we will learn to… > count the number of possible outcomes using a permutation
A permutation is an ordered arrangement of objects. The number of different
permutations of n objects is n!
5! = 5·4·3·2·1
3! = 3·2·1
2! = 2·1
The starting lineup for a baseball team consists of nine players.
How many different batting orders are possible using the starting
line up?
9! =
9·8·7·6·5·4·3·2·1 =
362,880
The teams in the National League Central Division are listed. How many different final standings are possible?
Chicago Cubs Cincinnati RedsHouston Astros Milwaukee BrewersPittsburgh Pirates St. Louis Cardinals
6! =6·5·4·3·2·1 = 720
The starting lineup for a baseball team consists of nine players. How many different batting orders are possible
from a group of 20 players?
__ · __ · __ · __ · __ · __ · __ · __ · __20 19 18 17 16 15 14 13 12
If you use 20!, how could you reduce out the 11 through 1?= 60,949,324,800
20! 11!
The number of permutations (ordered arrangements) of n different objects taken
r at a time is
nPr = n!
(n – r)!
9P6 = 9!
3!
Find 43P3 = 74,046
Find 26P10
= 19,275,223,970,000
43!40!
26!16!
43 · 42 · 41
26 · 25 · 24 · 23 · 22 · 21 · 20 · 19 · 18 · 17 =
Let’s Practice!!!!
Lesson 5.2.C Distinguishable
PermutationsIn this lesson, we will learn to…> count the number of possible outcomes using distinguishable permutations
> count the number of possible outcomes using combinations
B B G B G B B
What are Distinguishable Permutations?
Look at these arrangements…
B B B B BG G
Do you notice a difference?
How many ways can we order a group of 7?
If we have 4 A’s, 2 B’s, and 1 C, there are 105 distinguishable
permutations.
5040 ordered arrangements
7!
= 105
7! = 7·6·5·4·3·2·1
4 2
2!)(4!
The number of distinguishable permutations of n objects with A of one type,
B of another type, and so on is…
n!
A! · B! · C! · D! ···
A developer is planning a new subdivision that will consist of 6 one-story houses, 4 two-story
houses, and 2 split-level houses. In how many distinguishable ways
can the houses be arranged?
__! _! · _! · _!
= 13,86012
6 4 2
A contractor wants to plant 6 oak trees and 9 maple trees along
the street. If the trees are spaced evenly apart, in how many
distinguishable ways can they be planted?
__! _! · _!
= 5005156 9
What are Combinations?
Combinations tell us how many ways a GROUP of different things
can be formed.
Order does NOT matter.
With the letters A, B, C, and D there are 12 different ways to
select two letters.
PermutationsAB BA AC CA AD DABC CB BD DB CD DC
CombinationsBA AC CA AD DA
BC CB BD DB CD DCAB
You want to buy 3 CDs from a selection of 5 CDs. Label the CDs
A, B, C, D, and E. Write the 10 ways to make your selection.
ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE
Order doesn’t matter so ABC is the same as BAC.
Choose 3 CDs from a choice of 5 CDs
2 · 1
5 · 4 · 3 · 2 · 1
3!
because order doesn’t matter
combinations are like distinguishable permutations
=5 !
3 ! · 2 !
The number of combinations of r objects selected from a group
of n objects is…
n!
(n – r)! · r! nCr =
20!
(16! · 4!)20C4 =
The department of transportation receives 16 bids for a project. They plan to hire 4 companies.
How many different ways can 4 companies be selected from the
16 bidding companies?
__!__! · __!
= 182016
12 416C4 =
The manager of an accounting department wants to form a
3-person committee from the 16 employees in the department.
In how many ways can the manager do this?
__!__! · __!
= 56016
13 316C3 =
How many ways can a jury of 5 men and 4 women be selected from
12 men and 14 women?
(12C5)(14C4) =
792,792 ways
__!__! · __!
127 5
__!__! · __!
1410 4
Let’s Practice!!!!
A bag of M&M’s contains lilac, pink, and yellow candy. Construct a tree
diagram of the possible outcomes of picking 3 M&M’s out of the bag.
L P Y
L P Y
LPY LPY LPY
L P Y
LPY LPY LPY
L P Y
LPY LPY LPY
total possibilities
L P Y
L P Y
LPY LPY LPY
L P Y
LPY LPY LPY
L P Y
LPY LPY LPY
Find the probability of getting 2 yellow.
Sets with 2 yellow? 6
27= 0.222 = 22%
* ** ***
total possibilities
L P Y
L P Y
LPY LPY LPY
L P Y
LPY LPY LPY
L P Y
LPY LPY LPY
Find the probability of getting at least 1 lilac.
Sets with at least 1 lilac? 19
27= 0.704 = 70%
*** ****** **** * *** * *
total possibilities
L P Y
L P Y
LPY LPY LPY
L P Y
LPY LPY LPY
L P Y
LPY LPY LPY
Find the probability of getting 1 lilac, 1 yellow, and 1 pink.
Sets with one of each? 6
27= 0.222 = 22%
* * * * * *
In how many distinguishable ways can the letters in MISSISSIPPI be arranged?
__! __! · __! · __!
= 34,650114 4 2
If an M, 4 S’s, 4 I’s, and 2 P’s are randomly arranged, what is the probability that they will spell Missippippi?
1 34,650
=0.00002886= 0%
A test consists of 20 True-False questions. How many different ways are there to answer this test?
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 20 = 1,048,576How many of these ways are correct? 1
What is the probability of guessing and getting a 100 on the test?
11,048,576
= 0.000000954 = 0%
· · · · · · · · · · · · · · · · · · ·
Let’s Practice!!!!