Chapter 4 and 5 Recap

Probabilities

A summary : Chapter 4 and 5

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How do we feel about Chapter 4 and 5 in 1 to 2 words?

Quick
Definitions

T R Y T O D E F I N E T H E S E W O R D S B Y Y O U R S E L F

Probability

Event

Random Variable

Sample
Space S

Multiple Choice

What is a probability?

• How likely something is to happen.

• A type of mathematical equation
• A method of solving algebraic equations
• A branch of mathematics dealing with shapes and sizes

Multiple Choice

What is a sample space S?

• The set of all possible outcomes of a random experiment
• The space where samples are stored
• The space used for sampling
• The space where experiments are conducted

Multiple Choice

What is an event?

• A specific action or occurrence from the sample space

• A type of data structure from the sample space

• A programming language from the sample size

• A type of software used for sample space

Multiple Choice

What is a random variable in probability?

• A variable that can take on different values randomly based on outcomes

• A variable that is not affected by other variables
• A variable that always takes on a fixed value

Quick
Definitions

T R Y T O D E F I N E T H E S E W O R D S B Y Y O U R S E L F

Complementary

Events

Dependent vs
Independent

Events

Intersection
of Events

Union Of
Events

Mutually
Exclusive
Events

Match

Match the following definitions

• A and B cannot happen at the same time.

• A will occur if and only if B does not take place.

• The occurrence of A does not affect the occurrence of B.

• Consists of all outcomes that are in both A and B.

• Consists of all outcomes that are in A or in B or in both A and B.

Mutually Exclusive Event A and B

Complementary Events A and B

Independent Events A and B

Intersection of events A and B

Union of Events A and B

• Mutually Exclusive Event A and B

• Complementary Events A and B

• Independent Events A and B

• Intersection of events A and B

• Union of Events A and B

Complementary
Events

T H E P R O B A B I L I T Y O F “ A T L E A S T O N E ”

Categorize

Options (3)

Getting an even number, getting an odd number when rolling a die.

Getting a prime number, getting an even number when rolling a die.

All students attend class, no students attend class.

Are the events complementary?

Complementary

Not Complementary

Let's Talk about it!

Let's remember that
The complement occurs when the event doesn’t occur: If an event does not occur, then its complement occurs. If an event occurs, then its complement does not occur.

Let's Talk about it

a. Getting an even number, getting an odd number when rolling a die.

A number is either even or odd. There is no in-between. If a number is not even, it is odd, and vice versa.

b. Getting a prime number, getting an even number when rolling a die.

2 is a prime and even number. A number can be prime, and even the two events can happen at the same time, and one does not exclude the other.

c. All students attend class, no students attend class.

This one might look tricky, but just by using the definition, those two events are not complementary. If the "no students attend class" event does not happen, it does not mean all students will attend class.

Multiple Choice

If All students attend class is not the complementary event of no students attend class. then what is?

• Some students attend class
• All students are absent
• At least one student attends class

• Half of the students attend class

Categorize

Options (7)

Getting an even number, getting an odd number when rolling a die.

Getting a prime number, getting an even number when rolling a die.

All students attend class, no students attend class.

I will pass the exam, I will fail the exam.

It will rain tomorrow, it will be sunny tomorrow.

All members have different birthdays, two members have the same birthday.

All members have different birthdays, at least two members have the same birthday.

Let's Practice more

Are the events complementary?

Complementary

Not Complementary

Let's Talk about it

d. It will rain tomorrow, it will be sunny tomorrow.

Similarly to the one before, if the event "It will rain tomorrow" doesn't happen, it does not mean it will be sunny tomorrow. It may snow, be foggy, etc... Other events can occur.

e. I will pass the exam, I will fail the exam.

You either pass the test, or you fail. Failing is "NOT passing" Those two events are complementary.

f. All members have different birthdays, two members have the same birthday.

These two are not complementary. If the event "All members have different birthdays" does not happen, it doesn't necessarily mean that only two members have the same birthday.

e. All members have different birthdays, at least two have the same birthday.

If the event "All members have different birthdays" does not happen, it means that at least two members have the same birthday.

Multiple Choice

What is the complementary event of "It will be sunny tomorrow?"

• It will not be sunny tomorrow
• It will be rainy tomorrow
• It will be cloudy tomorrow
• It will be windy tomorrow

We will be using complementary events in all the upcoming chapters. So, let's make sure that we understand them correctly.

​

More Examples

More Examples

Multiple Choice

​The probability that I will cycle to work is .5.
The probability that I will take the train is .3 .
Determine the probability that I will neither cycle nor take the train.

• 0.2
• 0.3
• 0.5
• 0.8

Dependent vs Independent Events

C O N D I T I O N A L P R O B A B I L I T I E S

Intersection of Events :
A and B event

• Independent Events

• Dependent Events

A N D … D E N O T E D ∩ 𝑃 ( 𝐴 ∩ 𝐵 ) = 𝑃 ( 𝐵 ∩ 𝐴 )

The multiplication rule for Independent Events

Activity 3

The two spinners at the left are

spun. Find each probability.

• P(4 and A)

• P(less than 5 and B)

• P(even and C)

• P(Odd and A)

D E P E N D E N T V S I N D E P E N D E N T E V E N T S

Match

Match the following probabilities

• P(4 and A)

• P(less than 5 and B)

• P(even and C)

• P(Odd and A)

• $$\frac{1}{16}$$

• $$\frac{1}{6}$$

• $$\frac{1}{12}$$

• $$\frac{1}{4}$$

Let's Talk about it

Let's Talk about it

Union of Events :
A or B Event

O R - B O T H - E I T H E R … D E N O T E D U 𝑃 (𝐴 ∪ 𝐵 ) = 𝑃 (𝑩 ∪ 𝑨)

Mutually Exclusive
Events

Two events are mutually exclusive if

the events have no sample points in

common.

Multiple Choice

What is the probability that a student has a GPA between 2.0 and 3.0?

• 0.475

• 0.450

• 0.255

• 1

Multiple Choice

What is the probability that a student has a GPA under 2.0 and has skipped many classes?

• 0.314

• 0.313

• 0.080

• 0.68

Multiple Choice

What is the probability that a student has a GPA under 2.0 or has skipped many classes?

• 0.285

• 0.365

• 0.255

• 0.110

Let's Talk about it

Match

A survey of students to determine if they had a pierced ear was given. The results are summarized in the table to the left.

If one person is selected at random, find the probability that:

• They are Female given they are pierced

• They are Male given they are not pierced

• They are Not pierced given they are female

• They are Not pierced given they are male

0.89

0.82

0.1

0.8

• 0.89

• 0.82

• 0.1

• 0.8

Let's Talk about it

Multiple Choice

If P(A) = .2 and P(B) = .1, what is 𝑃(𝐴 and 𝐵) if A and B are independent?

• 0.02
• 0.3
• 0.1
• 0.2

Multiple Choice

If P(A) = .3 and P(B) = .4, what is 𝑃(𝐴 or 𝐵) if A and B are mutually exclusive?

• 0.7
• 0.12
• 0.5
• 1

Multiple Choice

If P(A) = .3 and P(B) = .4, what is 𝑃(𝐴 and 𝐵) if A and B are mutually exclusive?

• 0.3

• 0.12
• 0.4

• 0

Labelling

Events A and B are mutually exclusive. Suppose event A occurs with a probability of 0.39 and event B occurs with a probability of 0.52.

Drag labels to their correct position on the image

0.51

0.44

0.4

0.48

0.39

Since A and B are mutually exclusive :
P(A or B^c) = P(B^c) = 1-0.52
P(A or B^c) = 0.48

Let's Talk about it

​The phrase " A occurs, or B does not occur (or both)" refers to the occurrence of the event "A or B^c." On the Venn Diagram, we can see that Everything happens but B.

Since A and B are mutually exclusive :
P(A and B^c) = P(A) = 0.39
P(A and B^c) = 0.39

Let's Talk about it

The phrase "A either occurs without B occurring or A and B both occur" tells us that either A and B^c occurs or A and B occurs.
But we know that "A and B" is impossible because A and B are mutually exclusive. This is the very definition of mutually exclusive.
This reduces the phrase to A occurs without B: A and B^c

Multiple Choice

Among a group of boys, 70% like chocolate ice cream, 40% like strawberry ice cream, and 30% like both. If a boy is randomly selected from the group, what is the probability that he likes either chocolate or strawberry ice cream, but not both?

• 50%
• 60%
• 70%
• 80%

​Let's Talk About It

From the problem instructions
P(Chocolate) = 70%
P(Strawberry)=40%
P(Chocolate and Strawberry) = 30%
We are asked to find the probability that a boy likes only one flavor.
Answer = P(Chocolate OR Strawberry) - P(Chocolate and Strawberry)
P(Chocolate OR Strawberry) = P(Chocolate)+P(Strawberry) - P(Chocolate and Strawberry)
Hence, Answer = P(Chocolate)+P(Strawberry) - 2P(Chocolate and Strawberry) = 70% + 40% -2*30%
Answer = 50%

At least one Events

Let's say we toss a coin 3 times, what is the probability that head will come up at least once?

1. Let's define our sample space using a tree diagram

This is our sample space for tossing a coin three times.
For each time we toss, the probability of getting head is 0.5, and the probability of getting tail is 0.5.

​P(H) = 0.5
P(T) = 0.5

Sample Space

Fill in the Blanks

The complement of at least one is none.
In our case here, there are many ways we can get at least one head but only one way we do not get ANY heads.
That is if we get 3 tails.

P(At least One) = 1 - P(None)

P(At lease one head) = 1- P(No Head)
P(At lease one head) = 1- P(TTT)

At least one

Multiple Choice

What is the probability of getting at least one head from three coin tosses?

• 1-0.5³

• 0.5

• 0.5³

• 1+0.5²

Multiple Choice

Every day, Jorge buys a lottery ticket. Each ticket has a probability of 0.3

of winning a prize. After four days, what is the probability that Jorge has won at least one prize?

• 0.3

• 0.3⁴

• 1-0.7⁴

• 0.7⁴

Let's talk about it

1. What is the complement of winning at least one prize in 4 days?

2. What is the complement of winning a prize?

Drag and Drop

What is the complement of winning at least one prize in 4 days? ​

Placeholder

What is the complement of winning a prize?​

Placeholder

​

Drag these tiles and drop them in the correct blank above

Not winning a prize

Losing a prize

Not losing a prize

Not winning a prize in 4 days

Losing a prize in 4 days

Winning more than one prize in 4 days

Let's talk about it

1. What is the complement of winning at least one prize in 4 days?

   1. Not winning a prize in 4 days

2. What is the complement of winning a prize?

   1. Not winning a prize

Let's talk about it

1. What is the complement of winning a prize?

   Not winning a prize

What is the probability of not winning a prize ?

Recall : Each ticket has a probability of 0.3 of winning a prize.

Multiple Choice

What is the probability of not winning a prize ?

Recall : Each ticket has a probability of 0.3 of winning a prize.

• 0.5
• 0.3
• 0.7
• 1.0

Let's talk about it

1. What is the complement of winning at least one prize in 4 days?

   1. Not winning a prize in 4 days

P(Winning at least one prize) = 1 - P(Not winning a prize in 4 days)
What is the probability of not winning a prize in 4 days?

Multiple Choice

What is the probability of not winning a prize in 4 days?

• 4*0.7

• 0.7⁴

• 4^0.7

• 0.7

Multiple Choice

Every day, Jorge buys a lottery ticket. Each ticket has a probability of 0.3

of winning a prize. After four days, what is the probability that Jorge has won at least one prize?

• 0.3

• 0.3⁴

• 1-0.7⁴

• 0.7

Multiple Choice

The probability that a certain make of car will need repairs in the first six months is 0.8. A dealer sells six such cars. What is the probability that at least one of them will require repairs in the first six months? Round your final answer to four decimal places.

• 0.8858
• 0.9999

• 0.8888

• 0.0064

Let's talk about it

P( at least one car will require repairs in the first six months) = 1- P(No car will require repairs in the first six months)

P( at least one car will require repairs in the first six months) = 1- (1-0.8)⁶= 0.9999

Chapter 5

A probability distribution for a discrete random variable specifies the probability for each possible value of the random variable.

It is different from a probability model. The probability model lists the probability of each of those outcomes.

Categorize

Options (2)

Match the following

Probability Distribution

Probability Model

Multiple Choice

What is true about a true probability distribution?

• The sum of all probabilities is always greater than 1
• The sum of all probabilities is always less than 1
• The sum of all probabilities is always equal to 1
• The sum of all probabilities is always equal to 0

Multiple Choice

Suppose a factory that produces iPhones has a 5% defective rate. Use this information to create the charts below, then use the information to answer the following questions.

Event D : Iphone is defective

Event E : Iphone is not defective

Write out the sample space based on 3 iPhones being randomly selected:

• DDD, EDD, DED ,DDE ,EED ,EDE ,DEE ,EEE

• DDD, EED, DED ,DDE ,EED ,EDE ,DEE ,EDE

• DD,EED,DE,DD, EED ,EDE ,DEE ,EDE

• 3

Sample Space

Suppose a factory that produces iPhones has a 5% defective rate.

Event D : iPhone is defective

Event E : iPhone is not defective
We can use a tree diagram to display the outcomes that we may get.
Are events E and D independent?

Multiple Choice

Are events E and D independent?

• Yes, getting a defective iPhone in the first draw does not affect the next draw.

• No, getting a defective iPhone in the first draw does not affect the next draw.

• No, getting a defective iPhone in the first draw affects the next draw.

• Yes, getting a defective iPhone in the first draw affects the next draw.

Multiple Choice

Suppose a factory that produces iPhones has a 5% defective rate.

Event D: iPhone is defective

Event E: iPhone is not defective

3 iPhones are randomly selected:

What is the probability of Event E?

• 0.0451

• 0.95

• 0.000125
• 0.095

Multiple Choice

Suppose a factory that produces iPhones has a 5% defective rate.

Event D: iPhone is defective

Event E: iPhone is not defective

3 iPhones are randomly selected:

What is the probability of getting event EED?

• 0.0451

• 0.00125
• 0.000125
• 0.0135

Match

Suppose a factory that produces iPhones has a 5% defective rate.

Event D: iPhone is defective

Event E: iPhone is not defective

3 iPhones are randomly selected:

Match these events with their probabilities

• DDD

• DEE

• EEE

• DED

0.0001

0.0451

0.8574

0.0024

• 0.0001

• 0.0451

• 0.8574

• 0.0024

Probability Model

Suppose a factory that produces iPhones has a 5% defective rate.

Event D : iPhone is defective

Event E : iPhone is not defective
P(D) = 0.05
P(E) = 0.95

Probability Model

Suppose a factory that produces iPhones has a 5% defective rate.

Event D : iPhone is defective

Event E : iPhone is not defective
P(D) = 0.05
P(E) = 0.95

Match

Create the Probability Distribution based on 3 iPhones being randomly selected (Let x = # of defective iPhones):

• P(X=0)

• P(X =1)

• P(X=2)

• P(X=3)

0.8574

0.1354

0.0071

0.0001

• 0.8574

• 0.1354

• 0.0071

• 0.0001

Probability Distribution

​Create the Probability Distribution based on 3 iPhones being randomly selected (Let x = # of defective iPhones):

Multiple Choice

If three iPhones are randomly selected from the production line, what is the probability that all three are defective?

• 0.0001

• 0.1354

• 0.1426

• 0.0071

Multiple Choice

If three iPhones are randomly selected from the production line, what is the probability that at least one is defective?

• 0.0001

• 0.1354

• 0.1426

• 0.0071

Multiple Choice

If three iPhones are randomly selected from the production line, what is the probability that at least two are defective?

• 0.0001

• 0.1354

• 0.0072

• 0.0071

Multiple Choice

If three iPhones are randomly selected from the production line, what is the probability that exactly one is defective?

• 0.0001

• 0.1354

• 0.0073

• 0.0071