Mutually Exclusive Event.docx

Lesson Plan in Mathematics 10 I.

Weekly Objectives A. Content Standards The learner demonstrates understanding of key concepts of combinatorics and probability. B. Performance Standards The learner is able to use precise counting technique and probability in formulating conclusions and making decisions. C. Learning Competencies/Objectives 1. Illustrates events, and union and intersection of events using Venn diagram 2. Define the compound event and mutually exclusive events 3. Solve probabilities of union and intersection of event 4. Apply real life lesson involving union and intersection of events 5. Participate actively in class discussion

II. III.

Concept: Statistics and Probability Learning Resources A. References 1. Teacher’s Guide pages:279-292 2. Learner’s Materials pages: 330-340 3. Textbook pages:_________ 4. Additional Materials from Learning Resource (LR)portal: B. Other Learning Resources Internet, Google Procedures A. Preliminary Activities 1. Prayer 2. Greetings 3. Presentation of the Objectives of the Lesson

IV.

B. Establishing a purpose for the lesson Teacher’s Activity

Ok everyone kindly pass a ¼ sheet of paper with your names and date today. This will serve as our attendance.

Learner’s Activity

Every students inside the class will pass their ¼ sheet of paper containing their names.

C. Reviewing yesterday’s lesson or Presenting the new lesson

Last time we discuss how to get the probability of simple events.

The formula to find the probability of event A is given by:

Who can give me the formula to find the probability of event A ?

P(A) =

‘ Very Good ‘

number of element in A, and n(S) is the total number of sample space.

𝑛(𝐴) 𝑛(𝑆)

; where n(A) is the total

So for example:

To get the probability we use the formula

What is the probability of getting #3 when rolling a six sided die ?

‘Very good’

𝑛(𝐴)

1

P(A) = 𝑛(𝑆) OR P(3) = 6 Since in 6 sided die there just a single number 3, and there are 6 possible outcomes such as ( 1,2,3,4,5,6 )

How about the probability of picking a cards of heart in a deck of card? To get the probability we use the same formula 𝑛(𝐴)

P(A) = 𝑛(𝑆) OR P(cards of heart) =

‘Very Good ‘

So base on your own analyzation what can you say about Simple Events ?

13 52

Since there are 13 cards of heart in a deck of cards and there are 52 cards in 1 deck.

Base on my own opinion simple event is an event that have just only one outcome.

‘Very good’ Simple Event is any event which consists of a single outcome in the sample space.

D. Presenting examples/instances of the new lesson

Now that we already know how to solve the probability of simple events lets now discuss what is compound event. But before that let us recall some of your knowledge in Sets that you have already studied during your 1st year. How do you read this ? A U B

A UNION B

‘OK Very Good’ How about this one A ∩ B

A INTERSECTION B

Very Good This one ? A’ So who can illustrate those given sets using venn diagram ?

A Prime or not A or B compliment

Any one ? A U B A

B

Very good

A

Very good

∩

A

B

B

A’ Very good A It’s a good thing that you still know the concepts of set because at this point in time where going to study the probability of events using the concepts of sets. So lets go back to compound events Who among you have an idea what compound event is ? If a simple event is an event having a single outcome, how about compound event ?

For me, I think sir compound event is an event having more than just one outcomes.

Very good. So for in short a compound event consist of 2 or more simple events. For example; What is the probability of getting a #4 and #1 when rolling a 6 sided die ? Who wants to try to solve ? Ok very good.

1/6 + 1/6 = 2/6 or 1/3

How about what is the probability of getting a head and tail when tossing a coin? Its 0 sir, since there just only 2 outcomes in tossing a coin. So once the head happenthe tail is impossible to come up. That’s right! So that’s the compound events

How about mutually exclusive events? Any idea from the class? Ok that’s right meaning mutually exclusive events always have different outcomes. That’s why when one event happen it prevents another one to happen again.

Mutually exclusive event is an event that cannot occur both at the same time.

Like our previous example in tossing a coin. That’s is an example of mutually exclusive event since once the one outcome happen the other one willnot going to happen anymore. Another example; Let us say that our experiment is rolling a die. Event A is getting #4, and event B is getting an even numbers. So base on our definition do you think this is an example of mutually exclusive event or not ?

I think sir this is not an example of mutually exclusive event since the event A is getting #4 and the event B is getting an even numbers and as we all know 4 is an even number.

That right! Very Good. How about let say our experiment is drawing a ¼ paper that you passed a while ago. Then event A is getting a male, and event B is getting the paper of Bryan. Again base on our definition of mutually eclusive events do you think event A and B are mutually exclusive or not? But what if I let my event A is getting a female and event be remain the same. Getting the paper of bryan. Do you think clas this is an example of mutaully exclusive event or not ?

Still that is not an example of mutually exclusive event sir, because event A is getting a male and bryan is a male . so event A and B can happen both at the same time Yes it is! Because that 2 events cant happen both at the same time since bryan is a male and our event A is getting female.

So what is your observation about the events that is mutually exclusive and not mutually exclusive ? Anyone ?

When the 2 events have some elements in common in their sample space it is not mutually exclusive.

‘Ok very good ‘ If the 2 events have some intersections or even one, then thats enough to say that the events are not mutually exclusive. Understood ?

Yes sir..

Lets make some examples with solutions. Let say my experiments is rolling a die. Event A = ( 1,3 )

Event B = ( 2,4 )

Then waht is the probability of getting either A or B ? Who have some idea on how we can get that probabilty ?

I think sir we must solve first the probability of each events.

Ok thats right we must need to solve first the individual probability of the 2 events. So what is the probability of event A?

Using our formula for simple event , we obtain the probability of event A = 1/3 2

How about the probability of event B? Very good!

P(1,3) = 6 OR 1/3 Using the same formula it still 1/3 sir.. 2

P(2,4) = 6 OR 1/3 So wha tis the next step ? any one ? In the lesson of your set, when you here the word ‘or’ it refers to what ? Very good..

It means Union, So we must cpmbine all the elemets

So the P ( A U B ) = P(A) + P(B) On the board who wants to solve ?

P( A U B ) = P(A) + P(B) P(A U B) = 1/3 + 1/3 = 2/3

Veru good, since its Union of the 2 probabilities we just simply count the number of elements in event A and B then divide to total number of possible out comes. Who can illustarate it using a venn diagram ? Yess..

A

B

How about this example;

3

4

Same experiment rolling a die, But this time event A = ( 2,4,6 ) and event B = ( 4,5,6 )

1

2

Using the the steps we did a while ago who can solve it on the board ? Anyone ?

Performing the same steps we have; P(A) =

𝑁(𝐴) 𝑁(𝑆)

P( 2,4,6 ) = 3/6 OR ½ for event A P( 4,5,6 ) = 3/6 OR ½ for event B SO event A + event B = 1 So what have you observe to their probability ? Thats right!

It is 1 or 100%

But our elements in event is A = ( 2,4,6 ) and in event B = ( 4,5,6 ), Have you notice that?

Yes there have some error since it might be 1 or 3 comes up when you roll a die but in our event A and B there is no 1 and 3.

Thats a very good observation! So who can illustrate it using a venn diagram ?

A

B 2

4,6

5

Very good! Meaning in this example A and B are not mutually exclusive, as I have said a while ago if the events have even one intersection thats enough to say that the events are not mutually exclusive. What happened in this example is we count the probability of 4,6 twicw thats why the result become 100% which is wrong since it is possible to have 1 and 3. So what are we going to do to get the right answer?

Thats right! Then if the events are not mutually exclusive we are now going to use this formula P ( A U B ) = P(A) + P(B) – P (A ∩ B ) So that we’re not counting the probability of their intersection twice.

We must need to get rid of that intersections, so we need to subtract it.

E. Discussing new concepts and practicing new skills # 1

So in ½ cross wise answer the ff. Questions by pair.

1 If each of the 13. Outcomes in the sample space is equally likely. Find the probability of the ff. a. b. c. d.

1. a. b. c. d.

P(A) = ? P(B) = ? P( A or B ) = ? P( A and B) = ?

2. Classify each event as mutually exclusive or not. a. A card is drawn from standard deck. Event A – A face card is selected Event B – A diamond is selected b. 2 dice are thrown event A – The dice both show the same value. Event B – The sum of the numbers is 8 c. 2 dice are thrown event A – The dice both show the same value. Event B – The sum of the numbers is 9

4/13 5/13 8/13 1/13

2. a. NOT MUTUALLY EXCLUSIVE b. NOT MUTUALLY EXCLUSIVE c. MUTUALLY EXCLUSIVE

F. Discussing new concepts and practicing new skills # 2 It is often useful to use Venn diagram to visualize the probabilities of events. Study the Venn diagram and answer the questions that follow. (Do this by Pair) The extracurricular activities in which the senior class at Sta. Lucia High School participate are shown below.

Learner’s Answer: 1. The total number of the students in the Senior class is 345 2. The probability of those students who participated in the athletics is 159/345. 3. The probability that the student participates in athletics or drama is 227/345 4. The probability that the student participates only in drama and band is 30/345.

(See Activity 3 on page 332 of LM)

G. Discussing new concepts and practicing new skills # 2 Teacher will ask the following questions: 1. How were you able to find the total number of students in the senior class? 2. How does the concept of set help you in finding the intersection and union of two or more events? 3. What are some notations that are used in your study of sets in previous years that you can apply on these topics? How they are important in the study of probability of compound events? Since you know already what a union and intersection of event is, you are now ready for solving more problems on this concept.

Learners will answer: 1. By using the concept we have learned in Venn diagram. 2. The use of “or” means union of sets and the use of “and” means intersection of sets. 3. A U B means union of sets, while A Ռ B means intersection. They are important in the concept of joining or intersecting of events in the probability. To visualize easily the situation.

Learners will do the Activity individually.

Do Activity 7 on page 338 of your LM’s.

H. Developing mastery Teacher will let the students do Activity 8 on page 339 of LM. Discussion of answers will follow

Learners will do the Activity individually

I. Finding practical applications of concepts and skills in daily living

How would you use the concept of compound event in real life situation?

The concept of compound event can be used when there are two or more situations or problems that sometimes overlapped or just joined together. We need to see that whatever the circumstances are, there is always a solution to it.

J. Making generalizations and abstractions about the lesson

What is the formula for mutually exclusive events ? What is the formula if the events are not mutually exclusive.? What do you call to the events the consist of 2 or more simple events

P ( A U B ) = P(A) + P(B)

P ( A U B ) = P(A) + P(B) – P (A ∩ B ) Compound events

K. Evaluating learning A. Determine wether the following are mutually exclusive or not 1. P(A) = 1/8 P(B) = 2/5 P( A and B ) = 0 2. P(A) = 2/5 P(B) = 1/5 P( A or B ) = 3/5 3. P(A) = 0.3 P(B) = 0.4 P( A or B ) = 0.6 B. Use the given information to determine the probability of given events 1. P(A) = 3/5 P(B) = 1/3 A,B are mutually exclusive find P ( A or B ) 2. P(A) = 0.4 P(B) = 0.2 P ( A and B ) = 0.15 find P ( A or B ) 3. P(A) = 0.4 P(B) = 0.3 P ( A and B ) = 0.65 find P ( A and B ) L. Additional activities for application or remediation Do Activity 9 page 340 on Learner’s Module V.

Reflection I expect that there will be an 80% passing the assessment. There could be 20% learners who need additional activities for remediation. Remediation can be done

right after the Assessment to realize the outcome of 100% passing the remedial test. Remedial class will work. No need to have another remediation. The collaborative learning helped a lot in achieving all the objectives presented.

Prepared by: Mark Kevin M. Geradez ( Students Teacher ) Maria Ave B. Roque ( Critic Teacher )