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ProbabilityProbability
______________________________
_______________
ProbabilityProbability: is the : is the likelihood that an likelihood that an event will occur.event will occur. Probability can be written as a Probability can be written as a fraction or fraction or
decimal.decimal.
Probability is always Probability is always between 0 and 1between 0 and 1..
Probability = 0 means Probability = 0 means thatthat the event will the event will NEVER happen. NEVER happen. Example: Example: The probability that the Bills will win the The probability that the Bills will win the
Super Bowl this year.Super Bowl this year.
Probability = 1 meansProbability = 1 means the event will the event will ALWAYS happen.ALWAYS happen. Example: Example: The probability that Christmas will be on The probability that Christmas will be on
December 25December 25thth next year. next year.
Event:Event: A set of one or more outcomes Example: Getting a heads when you toss the
coin is the event Compliment of an Event: Compliment of an Event: The outcomes The outcomes
that are not the eventthat are not the event Example: Probability of rolling a 4 = 1/6. Not rolling Example: Probability of rolling a 4 = 1/6. Not rolling
a 4 = 5/6.a 4 = 5/6. Experiment: Experiment: aan activity involving chance,
such as rolling a cube Tossing a coin is theTossing a coin is the experiment experiment
Trial: Trial: Each repetition or observation of an experiment Each time you toss the coin is a Each time you toss the coin is a trialtrial
Outcome: Outcome: A possible result of an event.A possible result of an event. Example: An outcome for flipping a coin is Example: An outcome for flipping a coin is HH Example: The list of all the outcomes for flipping Example: The list of all the outcomes for flipping
a coin is a coin is {H, T} {H, T}
Sample space: Sample space: A list of A list of all the possible all the possible outcomes.outcomes. Example: The sample space for spinning the Example: The sample space for spinning the
spinner below is:spinner below is:
{B, B, B, R, R, R, R, R, Y, G, G, G}.{B, B, B, R, R, R, R, R, Y, G, G, G}.
Calculating OR Probabilities: Calculating OR Probabilities: by adding the by adding the probabilities.probabilities. Example: P (Red or Green) in the spinnerExample: P (Red or Green) in the spinner
Example: When rolling a die: P(3 or 4)Example: When rolling a die: P(3 or 4)
Uniform Probability: Uniform Probability: an event where all the an event where all the outcomes are equally likely.outcomes are equally likely. Which spinners have uniform probability?Which spinners have uniform probability?
3
2
12
8
12
3
12
5
3
1
6
2
6
1
6
1
Calculating Probabilities1. Rolling a 0 on a number cube
2. Rolling a number less than 3 on a number cube
3. Rolling an even number on a number cube
4. Rolling a number greater than 2 on a number cube
5. Rolling a number less than 7 on a number cube
6. Spinning red or green on a spinner that has 4 sections (1 red, 1green, 1 blue, 1 yellow)
0
3
1
6
2)12( or
2
1
6
3)6,4,2( or
3
2
6
4)6,5,4,3( or
16
6)6,5,4,3,2,1( or
2
1
4
2
4
1
4
1
Calculating Probabilities Contd
7. Drawing a black marble or a red marble from a bag that contains 4 white, 3 black, and 2 red marbles.
8. Choosing either a number less than 3 or a number greater than 12 from a set of cards numbered 1 – 20.
9
5
9
2
9
3
2
1
20
10
20
8
20
2
Independent Independent PracticePractice
Write impossible, unlikely, equally likely, likely, or certain It is _____________ to draw a striped pebble from the bag.
Drawing a white pebble from the bag is _________________.
Drawing a spotted pebble from the bag is _______________.
If you reach into the bag, it is ___________that you will draw a pebble.
You are _________ to draw a pebble that is not black from the bag.
What is the probability of not picking a black pebble from the bag above?
What is the probability of picking a spotted pebble from the bag?
likely
impossible
likelyequally _
unlikely
certain
6
1
12
2
3
1
12
4
Independent Practice Using a standard Deck of Cards, calculate the
following probabilities P(red)
P(7 of hearts)
**P(7 or a heart)
P(7 or 8)
P(a black heart)
P(face card)
2
1
52
26
52
1
13
4
52
16
52
1
52
13
52
4
13
2
52
8
13
3
52
12
52
0
Experimental vs Experimental vs Theoretical Theoretical ProbabilityProbability
Theoretical ProbabilityTheoretical Probability: the probability : the probability of of what should happenwhat should happen.. It’s based on a rule:It’s based on a rule: Example: Rolling a dice andExample: Rolling a dice and getting a 3 =getting a 3 =
Experimental Probability: Experimental Probability: is based on is based on an an experimentexperiment; what ; what actually happenedactually happened.. Example: Alexis rolls a strike in 4 out of 10 games. Example: Alexis rolls a strike in 4 out of 10 games.
The experimental probability that she will roll a strike The experimental probability that she will roll a strike in the first frame of the next game is: in the first frame of the next game is:
5
2
10
4
6
1
Theoretical vs Experimental Theoretical vs Experimental ProbabilityProbability
Experimental Experimental Fill in table:Fill in table:
What is the experimental What is the experimental probability of getting a red?probability of getting a red?
What is the experimental What is the experimental probability of getting a probability of getting a blue?blue?
What is the experimental What is the experimental probability of getting a probability of getting a yellow?yellow?
TheoreticalTheoretical Fill in Table:Fill in Table:
What is the theoretical What is the theoretical probability of getting a red?probability of getting a red?
What is the theoretical What is the theoretical probability of getting blue?probability of getting blue?
What is the theoretical What is the theoretical probability of getting a probability of getting a yellow?yellow?
Block Color
Frequency
Red
Blue
Yellow
Block Color
Frequency
Red
Blue
Yellow
Theoretical and experimental probability of an event may or may not be the same.
The more trials you perform, the closer you will get to the theoretical probability.
Try the FollowingTry the Following
Calculate and state whether they are Calculate and state whether they are experimental or theoretical probabilities.experimental or theoretical probabilities.
1. During football practice, Sam made 12 out of 15 field goals. What is the probability he will make the field goal on the next attempt?
2. Andy has 10 marbles in a bag. 6 are white and 4 are blue. Find the probability as a fraction, decimal, and percent of each of the following:
P(blue marble) b. P(white marble)
Experimental
Theoretical
15
12
5
2
10
4
5
3
10
6 %606.0 %404.0
3. If there are 12 boys and 13 girls in a class, what is the probability that a girl will be picked to write on the board?
4. Ms. Beauchamp’s student have taken out 85 books from the library. 35 of them were fiction. What is the probability that the next book checked out will be a fiction book?
25
13Theoretical
Experimental 17
7
85
35
5. What is the probability of getting a tail when flipping a coin?
6. Emma made 9 out of 15 foul shots during the first 3 quarters of her basketball game. What is the probability that the next time she takes a foul shot she will make it?
2
1Theoretical
Experimental 15
9
7. What is the probability of rolling a 4 on a die?
8. There are 8 black chips in a bag of 30 chips. What is the probability of picking a black chip from the bag?
6
1Theoretical
Theoretical15
4
30
8
9. Christina scored an A on 7 out of 10 tests. What is the probability she will score an A on her next test?
10. There are 2 small, 5 medium, and 3 large dogs in a yard. What is the probability that the first dog to come in the door is small?
10
7Experimental
Theoretical5
1
10
2
Predicting Predicting ProbabilitiesProbabilities
%
100
part
whole
part part
whole whole
Making PredictionsMaking Predictions: Remember for : Remember for predications we use predications we use proportions.proportions.
1. A potato chip factory rejected 2 out of 9 potatoes in an experiment. If there is a batch of 1200 potatoes going through the machine, how many potatoes are likely to be rejected?
2. Based on Colin’s baseball statistics, the probability that he will pitch a curveball is 1/4. If Colin throws 20 pitches, how many pitches most likely will be curveballs?
12009
2 x
204
1 x
267 potatoes
5 curveballs
3. If John flips a coin 210 times about how many time should he expect the coin to land on heads?
4. If the historical probability that it will rain in a two month period is 15%, how many days out of 60 could you expect it to rain?
5. If 3 out of every 15 memory cards are defective, how many could you expect to be defective if 1700 were produced in one day?
2102
1 x
170015
3 x
105 times
340 memory cards
60100
15 x 9 days
Compound EventsCompound Events
A Compound Event: A Compound Event: is an event that consists of is an event that consists of two two or more simple events.or more simple events. Example: Example: Rolling a die and tossing a coin.Rolling a die and tossing a coin.
To find the sample space of compound events To find the sample space of compound events we usewe use organized lists (tables) and tree diagrams. Example: A car can be purchased in blue, silver, red, or
purple. It also comes as a convertible or hardtop. Use a table AND a tree diagram to find the sample space for the different styles in which the car can be purchased.
The Fundamental Counting Principle (FCP): The Fundamental Counting Principle (FCP): a way to find a way to find all the possible outcomes of an all the possible outcomes of an event.event.
**Just multiply the number of ways **Just multiply the number of ways each event each event can occur.can occur.
Example: The counting principle for the car Example: The counting principle for the car purchase problem above:purchase problem above:
4 x 2 = 8 8 possible 4 x 2 = 8 8 possible outcomesoutcomes
‘‘And’ Events: And’ Events: This means to This means to multiply the multiply the events.events. Example: When flipping a coin and rolling a die:Example: When flipping a coin and rolling a die:
P (heads and 1)P (heads and 1)
P(T and odd)P(T and odd)
12
1
6
1*2
1
4
1
2
1*2
1
ExamplesExamples
1. Suppose you toss a quarter, a dime, and a nickel. What is the probability of getting three tails?
Make a tree diagram to show the sample space:
Use the FCP to check the total number of outcomes:2 x 2 x 2= 8 8 possible 2 x 2 x 2= 8 8 possible
outcomesoutcomes
2. A coin is tossed twice. What is the probability that you land on heads at least once?
Make a tree diagram to show the sample space
P (at least one H) = ¾ P (at least one H) = ¾
3. Find the probabilities of each of the following if you were to draw two cards from a 52-card deck, replacing the cards after you pick them.
P(Jack and 2)
P(Ace or 5)
P(King of hearts and red 2)
P(Jack and 14)
P(King or 12)
P(red Queen or 5)
169
1
2704
16
52
4*
52
4
13
2
52
8
52
4
52
4
1352
1
2704
2
52
2*
52
1
00*52
4
13
1
52
40
52
4
26
3
52
6
52
4
52
2
4. List the sample space for rolling two six-sided dice and their sums. Then calculate the following probabilities:
P(3 or 4)
P(at least one odd)
P(doubles)
P(1 and 6)
P(sum of 5)
P(sum of at most 4)
36
5
36
3
36
2
234567 12
119 108
3
4
4
55
5
66
6
6
77
77
7
88
1110
109
9
9
8
8
4
3
36
27
4
3
36
27
18
1
36
2
9
1
36
4
6
1
36
6
5. Peter has 6 sweatshirts, 4 pairs of jeans, and 3 pairs of shoes. How many different outfits can Peter make using one sweatshirt, one pair of jeans, and one pair of shoes?
A) 13 B) 36 C) 72 D) 144
6. For the lunch special at Nick’s Deli, customers can create their own sandwich by selecting one type of bread and one type of meat from the selection below.
In the space below, list all the possible sandwich combinations using 1 type of bread and 1 type of meat.
If Nick decides to add whole wheat bread as another option, how many possible sandwich combinations will there be?
WCWC RCRC
WRbWRb RRbRRb
66
Independent & Independent & Dependent EventsDependent Events
Suppose you have a bag of with 4 red, 5 blue & 9 yellow marbles in it.
From the first bag, you reach in and make a selection. You record the color and then drop the marble back into the bag. You repeat the experiment a second time.
This experiment involves a process called with replacement. You put the object back into the bag so that the number of marbles to choose from is the same for both draws. Independent Event.
Suppose you have a bag of with 4 red, 5 blue & 9 yellow marbles in it.
From the second bag you do exactly the same thing EXCEPT, after you select the first marble and record it's color, you do NOT put the marble back into the bag. You then select a second marble, just like the other experiment.
This experiment involves a process called without replacement You do not put the object back in the bag so that the number of marbles is one less than for the first draw. Dependent Event
As you might imagine, the probabilities for the two experiments
will not be the same.
An Independent Event: An Independent Event: is an event whose is an event whose outcome is outcome is notnot affected by another event. affected by another event. Example: Example: Rolling a die & flipping a coinRolling a die & flipping a coin With ReplacementWith Replacement
An Dependent Event: An Dependent Event: is an event whose is an event whose outcome outcome isis affected by a prior event affected by a prior event Example: Example: pulling two marbles out of a bag at the pulling two marbles out of a bag at the
same timesame time Without ReplacementWithout Replacement
“Is this problem with replacement?”OR
“Is this problem without replacement?”
Try the followingTry the following
A player is dealt two cards from a standard deck of A player is dealt two cards from a standard deck of 52 cards. What is the probability of getting a pair 52 cards. What is the probability of getting a pair of aces?of aces?
This is “This is “without replacementwithout replacement” because the player ” because the player was given two cardswas given two cards
P(Ace, then Ace) = P(Ace, then Ace) =
***There are four aces in a deck and you assume the first card is an ace.******There are four aces in a deck and you assume the first card is an ace.***221
1
2652
12
51
3
52
4
**Can cross cancel with **Can cross cancel with multiplication** multiplication**
A jar contains two red and five green marbles. A jar contains two red and five green marbles. A marble is drawn, its color noted and put back A marble is drawn, its color noted and put back in the jar. What is the probability that you in the jar. What is the probability that you select three green marbles?select three green marbles?
With replacementWith replacement
P(green, then green, then green) =P(green, then green, then green) = 343
125
7
5
7
5
7
5
What is the probability of rolling a die and getting an What is the probability of rolling a die and getting an even number on the first roll and an odd number on the even number on the first roll and an odd number on the second roll?second roll?
When flipping a coin and rolling a die, what is the When flipping a coin and rolling a die, what is the probability of a coin landing on heads and then rolling a probability of a coin landing on heads and then rolling a five on a number cube?five on a number cube?
A bag of candy contains 4 lemon heads and 5 war A bag of candy contains 4 lemon heads and 5 war heads. If Tim reaches in, takes one out and eats it, and heads. If Tim reaches in, takes one out and eats it, and then 20 minutes later selects another candy and its that then 20 minutes later selects another candy and its that as well, what is the probability that they were both as well, what is the probability that they were both lemon heads?lemon heads?
4
1
36
9
6
3
6
3
12
1
6
1
2
1
6
1
72
12
8
3
9
4
With replacement With replacement (independent)(independent)
With replacement With replacement (independent)(independent)
Without replacement Without replacement (dependent)(dependent)
Mary has 4 dimes, 3 quarters, and 7 nickels in Mary has 4 dimes, 3 quarters, and 7 nickels in her purse. She reaches in and pulls out a coin, her purse. She reaches in and pulls out a coin, only to have it slip form her fingers and fall back only to have it slip form her fingers and fall back into her purse. She then picks another coin. into her purse. She then picks another coin. What is the probability Mary picked a nickel both What is the probability Mary picked a nickel both tries?tries?
Michael has four oranges, seven bananas, and Michael has four oranges, seven bananas, and five apples in a fruit basket. If Michael picks a five apples in a fruit basket. If Michael picks a piece of fruit at random, find the probability that piece of fruit at random, find the probability that Michael picks two apples.Michael picks two apples.
12
1
240
20
15
4
16
5
4
1
196
49
14
7
14
7With replacement With replacement
(independent)(independent)
Without replacement Without replacement (dependent)(dependent)
A man goes to work long before sunrise every A man goes to work long before sunrise every morning and gets dressed in the dark. In his morning and gets dressed in the dark. In his sock drawer he has six black and eight blue sock drawer he has six black and eight blue socks. What is the probability that his first pick socks. What is the probability that his first pick was a black sock and his second pick was a blue was a black sock and his second pick was a blue sock?sock?
Sam has five $1 bills, three $10 bills, and two Sam has five $1 bills, three $10 bills, and two $20 bills in her wallet. She picks two bills at $20 bills in her wallet. She picks two bills at random. What is the probability of her picking random. What is the probability of her picking the two $20 bills?the two $20 bills?
91
24
182
48
13
8
14
6
30
1
90
3
9
1
10
2
Without replacement Without replacement (dependent)(dependent)
Without replacement Without replacement (dependent)(dependent)
A drawer contains 3 red paperclips, 4 green A drawer contains 3 red paperclips, 4 green paperclips, and 5 blue paperclips. One paperclip paperclips, and 5 blue paperclips. One paperclip is taken from the drawer and then replaced. is taken from the drawer and then replaced. Another paperclip is taken from the drawer. Another paperclip is taken from the drawer. What is the probability that the first paperclip is What is the probability that the first paperclip is red and the second paperclip is blue?red and the second paperclip is blue?
A bag contains 3 blue and 5 red marbles. Find A bag contains 3 blue and 5 red marbles. Find the probability of drawing 2 blue marbles in a the probability of drawing 2 blue marbles in a row without replacing the first marble.row without replacing the first marble.
48
5
12
5
12
3
28
3
7
2
4
3Without replacement Without replacement
(dependent)(dependent)
With replacement With replacement (independent)(independent)
SimulationsSimulations
An Simulation: An Simulation: is an experiment that is is an experiment that is designed to act out a give designed to act out a give event.event. Example: Use a calculator to simulate Example: Use a calculator to simulate rolling a rolling a
number cubenumber cube Simulations often use models to act out an Simulations often use models to act out an
event that would be impractical to perform.event that would be impractical to perform.
Try the followingTry the following
1. In football, many factors are used to evaluate how good a quarterback is. One important factor is the ability to complete passes. If a quarterback has a completion percent of 64%, he completes about 64 out of 100 passes he throws. What is the probability that he will complete at least 6 of 10 passes thrown? A simulation can help you estimate this probability…
a. In a set of random numbers, each number has the same probability of occurring, and no pattern can be used to predict the next number. Random numbers can be used to simulate events. Below is a set of 100 random digits.
b. Since the probability that the quarterback completes a pass is 64% (or 0.64), use the digits from the table to model the situation. The numbers 1-64 represent a completed pass and the numbers 65-00 represent an incomplete pass. Each group of 20 digits represents one trial.
c. In the first trial (the first row of the table) circle the completed passes.
d. How many passes were completed in this trial?
e. Continue using the chart to circle the completed passes. Based on this simulation what is the probability of completing at least 6 out of 10 passes?
66
66
88
77
77
44
55
55
66
7777
7/107/10
2. A cereal company is placing one of eight different trading cards in its boxes of cereal. If each card is equally likely to appear in a box of cereal, describe a model that could be used to simulate the cards you would find in fifteen boxes of cereal.
a. Choosing a method that has 8 possible outcomes, such as tossing 3 coins. Let each outcome represent a different card. For example, the outcome of all three coins landing on heads could simulate finding card #1.
b. Toss three coins to simulate the cards that might be in 15 boxes of cereal. How many times would you have to repeat?
15 times15 times
3. A restaurant is giving away 1 of 5 different toys with its children’s meals. If the toys are given out randomly, describe a model that could be used to simulate which toys would be given with 6 children’s meals.
Use a spinner with 5 equal Use a spinner with 5 equal sections, spin it 6 timessections, spin it 6 times
