Chapter 4 Probability Part 2

Joan DeRosa
15 Sept 201719:56
EducationalLearning
32 Likes 10 Comments

TLDRThis script explores various types of probability calculations, including the addition rule for mutually exclusive events, the subtraction rule for overlapping events, and the multiplication rule for independent events. It also delves into conditional probability and dependent events, providing examples such as blood type distribution, student surveys, and medical outcomes. The script aims to clarify the concepts by demonstrating how to calculate probabilities in different scenarios, emphasizing the importance of understanding the relationship between events.

Takeaways
  • πŸ“š Compound events are formed by combining two or more simple events using 'and' or 'or', with different calculations for mutually exclusive and overlapping events.
  • πŸ”’ For mutually exclusive events, the probability is found by adding the individual probabilities without overlap.
  • πŸ€” Overlapping events require the subtraction of the overlapping probability to avoid double counting when calculating the combined probability.
  • πŸ“Š An example given was a survey with responses categorized as favoring, opposing, or having no opinion on a policy change, illustrating the calculation of combined probabilities.
  • 🩸 A blood bank example showed how to calculate the probability of a donor having type B blood or Rh positive blood, including handling overlaps.
  • πŸ‘₯ The script explains how to handle relative frequencies and total populations in probability calculations when a total is not given.
  • 🌹 An example of independent events was given with Joanna selecting flowers from a vase, where the probability of selecting a rose twice is the product of individual probabilities.
  • ❀️ The multiplication rule for independent events was illustrated, showing how to calculate the probability of two events both occurring.
  • πŸ”„ Dependent events, where one event affects the probability of another, were explained, using an iPod giveaway as an example of 'without replacement'.
  • πŸŽ“ Conditional probability was defined as the probability of an event given that another event has already occurred, with examples from a hospital patient survey.
  • πŸ“‰ The script provided examples of calculating conditional probabilities, such as the likelihood of satisfaction with surgery outcomes based on the type of surgery performed.
Q & A
  • What is a compound event in probability theory?

    -A compound event in probability theory is an event that is formed by combining two or more simple events using the words 'and' or 'or'. It represents the occurrence of multiple events simultaneously or in sequence.

  • How is the probability of mutually exclusive events calculated?

    -The probability of mutually exclusive events is calculated by summing the individual probabilities of each event. Since these events cannot occur simultaneously, their combined probability is the sum of their individual probabilities.

  • What is the difference between mutually exclusive and overlapping events?

    -Mutually exclusive events are those that cannot happen at the same time, with no overlap in their probabilities. Overlapping events, on the other hand, can occur simultaneously, and there is an overlap in their probabilities which must be accounted for to avoid double-counting.

  • How do you calculate the probability of a random person surveyed opposing or having no opinion about a policy change?

    -To calculate this, you would find the individual probabilities of someone opposing the policy change and someone having no opinion, then add these probabilities together since these are mutually exclusive events.

  • What is the correct probability calculation for the survey example given in the script?

    -The correct calculation should be the sum of the probabilities of opposing (37/100) and having no opinion (36/100), which equals 0.73 or 73%.

  • How is the probability of a donor having type B blood or Rh positive blood calculated?

    -The probability is calculated by adding the individual probabilities of having type B blood and Rh positive blood, then subtracting the probability of the overlap (those who have both type B and Rh positive blood) to avoid double-counting.

  • What is the concept of independent events in probability?

    -Independent events are events where the occurrence of one event does not affect the probability of the occurrence of another event. The probability of both events happening is found by multiplying their individual probabilities.

  • How is the probability of dependent events calculated?

    -For dependent events, where the occurrence of one event affects the probability of the other, the calculation involves multiplying the probability of the first event by the conditional probability of the second event given the first.

  • What is conditional probability and how is it calculated?

    -Conditional probability is the probability of an event occurring given that another event has already occurred. It is calculated by dividing the probability of both events occurring by the probability of the event that is given.

  • Can you provide an example of calculating conditional probability from the script?

    -An example from the script is determining the probability that a person was satisfied with their surgery given that they had knee surgery. This would be calculated as the number of satisfied knee surgery patients over the total number of knee surgery patients.

  • What is the importance of understanding the difference between 'with replacement' and 'without replacement' in probability problems?

    -Understanding 'with replacement' and 'without replacement' is crucial because it affects the total number of outcomes and thus the probability calculation. 'With replacement' means the total number of outcomes remains the same after each event, while 'without replacement' means the total decreases as events occur.

Outlines
00:00
πŸ“Š Understanding Compound Event Probabilities

This paragraph explains the concept of compound event probabilities, which are calculated by combining the probabilities of two or more simple events. It distinguishes between mutually exclusive events, where the probability is the sum of individual probabilities, and overlapping events, where the overlap must be subtracted to avoid double-counting. An example using a survey about change and policy is provided, illustrating how to calculate the probability of a respondent opposing or having no opinion on the matter. The summary also corrects a mistake in the calculation, emphasizing the importance of accurate arithmetic in probability assessment.

05:01
🧬 Probabilities with Overlapping and Independent Events

The second paragraph delves into the calculation of probabilities involving overlapping and independent events. It uses the example of a blood bank cataloging blood types to demonstrate how to account for overlaps when calculating the probability of a donor having type B blood or being Rh positive. The paragraph also discusses the multiplication rule for independent events, as illustrated by the probability of selecting a rose twice from a vase in a series of random selections. The summary corrects the probabilities provided in the script and emphasizes the importance of understanding the relationship between events when calculating probabilities.

10:03
πŸ”„ Dependent Events and Conditional Probability

This paragraph introduces the concepts of dependent events and conditional probability. It explains that dependent events occur when the occurrence of one event affects the probability of another, often in scenarios without replacement. The paragraph provides an example of an iPod giveaway to illustrate the calculation of dependent probabilities. It also explains conditional probability, which is the probability of an event given that another event has already occurred, using a survey of surgery patients as an example. The summary highlights the importance of considering the total outcomes based on given information when calculating conditional probabilities.

15:06
πŸ“‰ Calculating Conditional Probabilities in Surveys

The final paragraph focuses on calculating conditional probabilities using data from a survey of surgery patients. It demonstrates how to determine the probability of satisfaction or dissatisfaction with surgery results based on specific types of surgery. The paragraph emphasizes the importance of using the correct totals and given information to calculate accurate probabilities. The summary provides corrected probabilities and illustrates the step-by-step process of calculating conditional probabilities in the context of survey results.

Mindmap
Example: Surgery Patient Survey
Given Event Basis
Example: iPod Giveaway
Without Replacement
Conditional Probability
Example: Flower Selection
Multiplication Rule
Example: Blood Bank Donors
Example: Survey on Change and Policy
Overlapping Events
Mutually Exclusive Events
Conditional Probability
Dependent Events
Independent Events
Relative Frequency and Totals
Compound Events
Probability Concepts
Alert
Keywords
πŸ’‘Compound Event
A compound event in probability theory is an event that can occur in more than one way. It is composed of two or more simple events. In the video, compound events are discussed in the context of using 'and' or 'or' to combine simple events, emphasizing the difference in calculation when events are mutually exclusive versus overlapping.
πŸ’‘Mutually Exclusive Events
Mutually exclusive events are events that cannot occur at the same time. The script explains that when calculating the probability of such events, you simply sum the individual probabilities because there is no overlap. An example from the script is finding the probability of a person being opposed or having no opinion about a policy change, where these two outcomes do not overlap.
πŸ’‘Overlapping Events
Overlapping events are those that can occur at the same time, meaning they share some outcomes. The script discusses the necessity to account for this overlap when calculating probabilities to avoid double-counting. For instance, the probability of a blood donor having type B blood or being Rh positive involves subtracting the overlap (those who are both type B and Rh positive) from the total count.
πŸ’‘Relative Frequency
Relative frequency is a measure used to estimate the probability of an event by the number of times the event occurs relative to the total number of trials. The script mentions this concept in the context of finding probabilities when a total number is not given, requiring the calculation of a total from the given frequencies.
πŸ’‘Independent Events
Independent events are those where the occurrence of one event does not affect the probability of the occurrence of another. The video explains that the probability of two independent events both occurring is found by multiplying their individual probabilities, as in the example of selecting a rose twice from a vase with replacement.
πŸ’‘Dependent Events
Dependent events are those where the occurrence of one event affects the probability of the occurrence of another. The script illustrates this with an example of choosing iPods from a bag without replacement, where the probability of choosing an orange iPod changes after the first selection.
πŸ’‘Conditional Probability
Conditional probability is the probability of an event given that another event has already occurred. The video script explains this by using examples such as the probability of a patient being satisfied with surgery given that they had knee surgery, where the total number of outcomes is reduced to only those relevant to the condition (knee surgery).
πŸ’‘Without Replacement
The term 'without replacement' refers to a scenario where an item is removed from a set and not put back, thus reducing the total number of items for subsequent selections. The script uses this in the context of dependent events, such as choosing iPods from a bag, where the selection of one iPod affects the probabilities of future choices.
πŸ’‘Probability Calculation
Probability calculation is the process of determining the likelihood of an event occurring. The video script covers various methods of calculating probabilities, such as adding individual probabilities for mutually exclusive events, adjusting for overlaps in non-exclusive events, and using multiplication for independent events.
πŸ’‘Survey Data
Survey data is information collected from individuals through questionnaires or polls. The script uses survey data to illustrate examples of calculating probabilities, such as determining the probability of a student being a senior or reading a daily paper based on the responses of surveyed students.
Highlights

Introduction to compound events and their calculation using 'and' or 'or'.

Explanation of how to calculate probabilities for mutually exclusive events by summing individual probabilities.

Clarification on handling overlapping events in probability calculation by subtracting the overlap to avoid double counting.

Example of calculating the probability of a respondent opposing or having no opinion on policy change from a survey.

Illustration of probability calculation for blood type B or Rh positive using a blood bank's data, including handling overlaps.

Misinterpretation correction: The correct probability calculation for opposed or no opinion should result in 0.55, not 0.73.

Demonstration of how to find the total and calculate relative frequency when a total is not provided.

Use of a survey to determine the probability of students being seniors or reading a daily paper, accounting for overlap.

Introduction to the multiplication rule for calculating the probability of independent events occurring together.

Example of calculating the probability of selecting a rose twice in a row from a vase of flowers.

Misinterpretation correction: The correct probability for selecting a rose twice should be approximately 0.16.

Explanation of calculating the probability of dependent events, where the outcome of one event affects the other.

Example of calculating the probability of choosing an orange iPod by two people without replacement.

Introduction to conditional probability and its calculation based on given events.

Example of determining the probability of satisfaction with surgery results given that the surgery was on the knee.

Misinterpretation correction: The correct probability for dissatisfied patients with hip surgery should be approximately 0.143.

Example of calculating the probability of a patient having heart surgery given dissatisfaction with the surgery results.

Transcripts
00:00

cow pal probabilities a compound event

00:04

combines two or more simple events using

00:09

the word an or the word or when we're

00:14

working with mutually exclusive events

00:17

the probability is found by summing the

00:20

individual probabilities of the events

00:23

so if we have a probability of A or B

00:27

what you would do to find the

00:29

probability is find the probability of a

00:32

and then find the probability of B and

00:35

add them together when we have

00:39

overlapping events the events are not

00:43

mutually exclusive so mutually exclusive

00:46

has no overlap in the probability but

00:50

when we have overlapping events the

00:53

probability that overlapping events a

00:56

and B or both will occur is expressed as

01:00

so the probability of a or the

01:03

probability of B would equal you would

01:07

find the probability of a you would add

01:09

it to the probability of B but then you

01:12

would need to subtract out the

01:14

probability of that overlap so whatever

01:18

the overlap is so you're not including

01:21

that value twice so you always have to

01:25

subtract out the overlap so let's do an

01:27

example here in a survey about change

01:31

and policy a hundred people were asked

01:33

if they favor it opposed or had no

01:36

opinion so you see the values in the

01:39

table that we have here now we're asked

01:42

to find the probability that a randomly

01:44

selected responded to the survey opposed

01:48

or had no opinion about the change of

01:52

policy so we're looking at the opposed

01:55

or no opinion so we're finding that that

01:59

that the probability is mutually

02:02

exclusive because there's no overlap in

02:05

these events so when I find the

02:07

probability I'm going to find the

02:09

probability of a pose

02:11

and a probability of no opinion and I'm

02:14

going to add them together so a

02:17

probability of opposed would be that 37

02:20

over a hundred plus the probability of

02:23

no opinion would be 36 over a hundred so

02:28

when I add this together I end up

02:29

getting 73 over a hundred or it would

02:33

equal point seven three or 73% pause and

02:39

try so in this case you should have

02:43

ended up with 0.55 as the probability

02:47

pause and try and in this case you would

02:52

end up with 0.35 as your probability

02:56

pause and try now remember when you have

03:02

a relative frequency if you don't have a

03:04

total you're going to have to find the

03:06

total and in this case the total was 36

03:09

and when you find the probability you

03:12

end up with point four four four so

03:17

let's do an example here a blood bank

03:19

catalogs the type of blood given by

03:22

donors during the last five days a donor

03:25

is selected at random we want to find

03:28

the probability the donor has type B or

03:31

is Rh positive so what we're looking at

03:35

this table when we look at type B the

03:38

type B is a total of forty five out of

03:42

the 409 and the positive or the H of the

03:48

Rh positive we have a total of three

03:52

hundred and forty four out of the 409

03:55

that have positive blood type so when we

03:59

look at this you can see that we have an

04:01

overlap of thirty seven meaning that the

04:06

type B the 37 was added in the total for

04:10

forty five and for the positive that 37

04:14

was added into the total of three forty

04:17

four so because it's added in both

04:20

totals you cannot include it in both of

04:23

your profit

04:24

oh these so when you're doing the

04:26

probability you're going to need to take

04:28

out the overlap the the doubling of that

04:31

37 so how to do this you want to find

04:35

the probability of B first which would

04:38

be that 45 over 409 and then you're

04:42

going to find the probability of

04:43

positive which would be that 344 over

04:47

409 and then you want to subtract out

04:50

one of those 37 because you used it

04:53

twice so you only need to subtract out

04:55

one so you're going to subtract out the

04:57

37 over 409 to find the actual

05:01

probability of this so this is that

05:04

overlap that needs to be subtracted out

05:06

so your probability for being a type B

05:10

or Rh positive would be point eight six

05:14

one pause and try so in this case you

05:21

should have gotten a probability of 0.8

05:24

four zero pause and try so in this case

05:31

you should have gotten a probability of

05:33

0.85 one pause and try so this

05:40

probability is point six nine three

05:44

pause and try and this probability is

05:49

point nine nine five this next example

05:53

is we don't have a table so we can't see

05:57

the overlap we need to define what the

06:00

overlap is so of fifteen hundred and

06:04

sixty students surveyed 840 were seniors

06:08

and six thirty read a daily paper the

06:13

rest of the students were juniors only

06:16

two hundred fifteen of the papers read

06:19

were juniors now we're asked to find the

06:22

probability that a student was a senior

06:26

or read the daily paper read a daily

06:29

paper so when you're looking at this we

06:33

want the probability of a senior so we

06:36

can find that easily

06:38

now when we find the order and the

06:41

Orbeez we're adding the readers the

06:44

probability of readers we have this six

06:48

thirty over the fifteen sixty but you

06:52

have to be careful here because we have

06:55

an overlap and the overlap is the

06:58

seniors that read the paper the daily

07:02

paper so how to find the senior total

07:06

that read the daily paper is you have to

07:08

go by the information that's given and

07:11

the information here that's given is

07:13

we're told that out of those six hundred

07:17

and thirty readers two hundred fifteen

07:20

of them are the juniors so to find the

07:23

seniors we need to take the six thirty

07:26

and subtract out the juniors so the

07:29

total for the readers for that were

07:31

seniors were that four hundred and

07:34

fifteen so that's my overlap here that

07:38

415 seniors that read and I need to take

07:42

that overlap out and then for in order

07:45

to find this true probability so we end

07:48

up with 0.67 6 as our probability pause

07:56

and try so this is similar here where we

08:02

have a probability where a hundred sixty

08:06

beauty spas customers were surveyed and

08:10

ninety six had hairstyle and 61 had

08:15

manicures and you see here they're

08:17

telling you the twenty eight of the

08:19

customers only had manicures so that is

08:23

an overlap when we talk about the

08:26

combination of hairstyle and manicures

08:30

so you need to take the 61 and subtract

08:34

the twenty eight to find the overlap of

08:37

people who had the hair style and

08:40

manicure which is that 33 so you should

08:44

have gotten a probability of point seven

08:47

seven five

08:50

pause and try so again that overlap is

08:56

that 52 and you should have gotten a

08:58

probability of 0.6 1 so the next type of

09:04

compound probability when we have

09:07

independent events and two events a and

09:11

B are independent if a occurs and it

09:14

doesn't affect the probability of B

09:17

we're going to be using the

09:19

multiplication rule for the probability

09:21

of a and B happening so when we're

09:25

talking about the probability of a and B

09:28

we're going to be multiplying and you

09:30

see you're going to multiply the

09:32

probability of A to the multiply to the

09:35

probability of B so we have an example

09:39

here where Joanna has three roses for

09:43

tulips and one carnation in a vase she's

09:46

gonna randomly select one flower she

09:49

took up photo and took a photo of it and

09:52

put it back

09:54

she then repeats these steps what is the

09:58

probability she selected a rose both

10:00

times so you want to find the

10:03

probability of her selecting a rose both

10:06

times and because she put the first

10:10

flower back that doesn't change the

10:14

total when she selects the second rose

10:18

so when you're doing the probability

10:21

here you're going to find the

10:23

probability of it being a rose the first

10:25

time and then you're going to find the

10:27

probability of it being a rose the

10:29

second time so when you do the

10:31

probability here you're going to

10:32

multiply the probability of a rose which

10:35

is three out of eight times the

10:38

probability of a the second one being a

10:40

rose which would be three out of eight

10:42

and when you get this probability you're

10:45

going to end up with approximately point

10:48

four point one at four one pause and try

10:55

so in this case you should have gotten a

10:58

probability of 0.16

11:03

pause and try so this one you're given

11:08

the probabilities already it's a 90

11:10

percent chance of survival so we have

11:12

that and the patient is 45 percent

11:16

chance of heart damage you're given that

11:18

so when you're finding the probability

11:21

of survival and heart damage heals

11:25

you're going to multiply the two

11:27

probabilities together you say you

11:29

should have ended up with point four

11:31

zero five pause and try so in this case

11:38

you're going to multiply the point eight

11:41

five to itself three times and you get

11:44

point six one

11:47

so now this next one is when we have

11:50

dependent events two events a and B are

11:54

dependent if a occurs and it affects the

11:57

probability of B occurring so dependent

12:00

will affect the second probability so a

12:04

probability of a and B occurring we

12:08

would end up having the probability of a

12:10

times the probability of a given that a

12:14

already happened so a lot of times you

12:17

might not have you might have in there

12:20

the words without replacement so if

12:24

you're taking something out then the

12:27

total is going to change if it says it's

12:30

without replacement so a key note here

12:35

sometimes a problem will not

12:37

specifically state whether it is a

12:40

problem with or without replacement but

12:43

you have to use your own common sense

12:46

when it comes to a probability problem

12:51

if we're talking about people maybe

12:54

going on a trip and you're going to

12:56

choose five of your friends we can't and

12:59

you're going to choose two out of the

13:01

five we can't choose the same friend

13:04

twice so it will be a dependent event so

13:09

let's do this example here Best Buy is

13:12

having an iPod giveaway they put all

13:16

the iPod shuffles in a bag customers may

13:19

choose an iPod without looking at the

13:22

color inside the bag there are four

13:24

orange five blue six green and five pink

13:28

iPods if Maria chooses one iPod at

13:33

random and that her sister chooses one

13:35

PI iPod at random what's the probability

13:38

they are both choosing an orange iPod so

13:42

again the first thing you're going to

13:43

need here is you're going to need some

13:45

type of total so you're going to have to

13:47

actually add four plus five plus six

13:50

plus five together you need your totals

13:52

now when we get that total we want to

13:55

find the probability of orange Maria

13:57

getting an orange and then the

13:59

probability of her sister getting an

14:01

orange now in this case it's not telling

14:04

me whether it's replaced or not though

14:07

if she if Maria is choosing an iPod

14:10

she's going to keep it she's not going

14:13

to put it back in so when you're finding

14:15

the probability you're going to end up

14:17

with Maria's probability is going to be

14:19

four out of 20 but because Maria's not

14:22

given our iPod back her sister is going

14:25

to choose an iPod and it's going to

14:27

change the value of the probability if

14:31

Maria has a orange iPod then there's

14:35

only three oranges left and the total

14:38

number of iPods change because Maria

14:40

kept her iPod so you end up having three

14:44

over nineteen as her sister's

14:48

probability and when you multiply this

14:50

together you should get a probability of

14:53

approximately point zero three two so

14:57

again when you have a probability less

14:59

than point zero five it's unlikely

15:02

something like this would happen

15:06

pause and try so in this case you should

15:11

have gotten point four six seven pause

15:15

and try

15:18

so in this case you should have gotten

15:21

point two eight six pause and try so in

15:29

this case you should have gotten approx

15:31

point two nine four the next type of

15:36

probability we're going to work with is

15:37

a conditional probability and a

15:40

conditional probability is the

15:42

probability of an event occurring given

15:45

that another another event has already

15:47

occurred okay so the probability of B

15:52

given a so you see how it would be

15:54

written in the probability denoted a key

16:00

note here is that the total outcome is

16:03

always going to be based on the given so

16:06

when we're looking at probabilities

16:08

remember the probability is based on

16:11

totals so you need a total but in a

16:14

conditional probability the total is

16:17

going to be based on the given because

16:20

we're already told something and that

16:22

eliminates everything else that is not

16:25

part of that given information so let's

16:30

do an example here survey given to

16:33

surgery patients at a given Hospital

16:35

results are displayed in the table below

16:39

determine the probability that the

16:41

person was satisfied with the results of

16:44

their surgery given that the person had

16:48

knee surgery so we already know that the

16:51

person had knee surgery so because we

16:54

know that our total is going to be based

16:58

on the fact that they've knee surgery so

17:01

the total in this probability is going

17:03

to be 95 that is the net total number of

17:08

knee surgery everything else doesn't

17:10

matter because we already know that it's

17:13

based on the fact they had knee surgery

17:15

and that they were satisfied is going to

17:19

be the seventy in the satisfied for knee

17:23

surgery so this probability here would

17:26

be 70 over 95 or approximately point

17:31

seven

17:32

3:7 so we're using the same example but

17:37

we're asked determine the probability

17:40

was dissatisfied with the results of the

17:42

survey surgery given that the person had

17:47

hip surgery so now the given is based on

17:50

the hip surgery and the hip surgery

17:53

total is that fought 105 so because the

17:57

given is the hip surgery we know that

17:59

it's in hip surgery the dissatisfied is

18:02

that 15 so then we end up with 15 over

18:07

105 which is approximately 0.14 3 and

18:13

this next example you see the same

18:17

determine the probability that the

18:20

person had heart surgery given that the

18:23

person was dissatisfied with the results

18:26

of the survey or surgery so again

18:30

dissatisfied is the total of 45 so we

18:34

need to use the total of dissatisfied

18:37

surgeries and then we look at what we're

18:41

looking for the probability so it would

18:43

be 5 divided by 45 which is

18:46

approximately 0.1 1 pause and try so in

18:53

this case you should have gotten point 3

18:55

6 0 pause and try so in this case you

19:02

should have gotten 0.375 pause and try

19:07

so here we should have gotten point 3 1

19:11

6 pause and try this you should have

19:16

gotten point 6 2 5 pause and try this

19:23

you should have gotten point 2 2 2 pause

19:28

and try

19:30

this you should have gotten point 796

19:34

pause and try this you should have

19:39

gotten point one seven eight pause and

19:43

try this you should have gotten point

19:47

one five four pause and try and this you

19:53

should have gotten point four two nine