Probability Formula Sheet: bit.ly/3zb22rW Final Exams and Video Playlists: www.video-tutor.net/ Full-Length Videos & Worksheets: www.patreon.com/MathScienceTutor/collections Next Video: ruclips.net/video/lWAdPyvm400/видео.html
Good day Sir. Thank you for all you do. Your channel have been of immense help to me various ways, more help than my school lecturers ever were. Thanks so much, may God multiply the blessings in your blessed life on my behalf, Amen. Please Sir, I need your clarification on question 2 B 12:18 . To me, you are a demi maths god but I beg to disagree with you on it. In my thought, it's either you want to test if we are paying detailed attention to the formula or you got it wrong. From what I have learnt from you and what I know in probability, the solution should be as follows: P(Br|M) = 70/90 = 7/9 Or P(Br|M) = P(M and Br)/P(M) =(70/200)/(90/200) = 70/90 = 7/9 = 0.7778 = 77.78%. Please correct and clarify me if I am wrong. Please tag me when you do. Thanks
Dude, your videos are always so clear and to the point. You're a godsend. Crazy how it doesn't even require any fancy animations or anything, just clear examples and explanations. You are a fantastic teacher!
Omg THANK YOU SO MUCH FOR THIS LESSON. I literally sat in my math class for an hour and 40 min straight and I can't understand anything that my teacher is saying. Literally your video is less than 20 min and I understood right away!
I don't know why I pay tuition fees for my university faculty. If i don't understand anything at the end of the day, i have to come to this RUclips channel and make a solution. And things are very easy to understand. If The Organic Chemistry Tutor had been my university faculty, I probably wouldn't have had to struggle so much. May Allah grant you long life❤️❤️❤️❤️
Me spending math online class for 2 periods 😶 vs me spending math class with this guy 😲. . . . . . . . . . . . . . . THANK YOU FOR MAKING MY LIFE EASIER!!! I DON'T UNDERSTAND ANYTHING THAT I LEARNT DURING ONLINE CLASSES 😭
Thank you very much for your video. Now I clearly understood conditional probability with Venn diagram amd Contingency table. Once again I am so glad and thank you very much
OMG BETTER THAN MY MATH TEACHER. SHE HAD NO IDEA HOW TO EXPLAIN THE FORMULA AND OUR TEST IS TOMORROW, AND YOU LITERALLY JUST SAVED MY REPORT CARD.. TYSM!!!
Thanks a lot for comments like this. I thought what I knew was wrong. I had just taught that part to someone and was alarmed when I saw the vid that I might have taught it wrong. Phew.
A = 150. A and not B = 120. I wanted to clarify that, but yes, you are right that in a Venn Diagram, we usually split the values by shaded regions, and thus the "A and NOT B" region should have had 120. While A = A and NOT B + A and B = 150.
It proved to be an important point of discussion. I was thinking that same. 150 is the number of only Algebra. 180 is the number of Algebra students total.
I love your explanations and ypur channel. But I think in the 1st example you should from 150 substract 30, because these 150 include All students that have Algebra and they also may have Chemistry. and 150-30 it will be 120 students that take Only Algebra. What do you think about it?
The picture isn't drawn quite clear, it'd probably better to put numbers like 120/30/50 i.e. labelling the mutually disjoint parts, I believe it's more intuitive and established notation if we put numbers on Euler-Venn diagrams. However "the student is taking Algebra" should not be interpreted the exclusive "Only Algebra" so that calculation is correct, whereas if we question the probability of Only Algebra => it will be 120/500 for sure.
This is a fantastic explanation! So can we think of conditional probability as 'changing' the sample space to a new base? For example if we have a 'given it has brown eyes' the new sample space is the column Br and we have to compute any other conditional probability by referring back to this sample space instead of the global sample space.
One thing that bugs me: If 150 are in algebra, and 30 are in algebra AND chemistry, then wouldnt the probability of selecting a student in algebra be 180/500 = 0.36 ? Then the P(C|A) = 30/180.
Your such a good teacher. I was confused though about min. 3:15 because you said divided by P(B) however in 6:25 you say P(A). Was just confused about that.
good explanation. however, I have a question what is the difference between: a female with brown eyes and a female given that it has brown eyes? I see they are the same? why have choosen a conditional probablities while I see it is a simple question of a female with brown eyes which should be 100/200 = 0.5 (50%) at 13:40
I think you made an error with the Venn diagram. You should've subtracted the 30 from 150 and 80. Your diagram doesn't seem to represent what the question says.
Good day Sir. Thank you for all you do. Your channel have been of immense help to me various ways, more help than my school lecturers ever were. Thanks so much, may God multiply the blessings in your blessed life on my behalf, Amen. Please Sir, I need your clarification on question 2 B 12:18 . To me, you are a demi maths god but I beg to disagree with you on it. In my thought, it's either you want to test if we are paying detailed attention to the formula or you got it wrong. From what I have learnt from you and what I know in probability, the solution should be as follows: P(Br|M) = 70/90 = 7/9 Or P(Br|M) = P(M and Br)/P(M) =(70/200)/(90/200) = 70/90 = 7/9 = 0.7778 = 77.78%. Please whoever can should correct and clarify me if I am wrong. Please whoever does should tag me when he/she does. Thanks
I don't get something. At (12:16) you compute a union probability, which can be calculated by multiplying the individual probabilities of the events right? Thus, Male bird (90/200) * brown eyes (170/200) = 0.38, not 0.35.
I think you did one step wrong, I might be wrong tho. In the question with the students taken Algebra and Chemistry the probability of P(A) should be 180/500 ya? not 150/500. This is because the 30 students also took Algebra as well
Quick question about the algebra and chemistry word problem... Since there are 30 students taking both the classes, shouldn't the number of students taking only algebra be 120 students and only chemistry be 50 students?
Arvind Chandrashekar no, the intersection simply means how many students are in both A and C. It was stated in the problem that A has 150 students, C has 80 students, and 30 of those students are in both A and C. The way the venn diagram is a little counter intuitive, but pay attention to what the problem statement says. These things were given to us, so we assume they are true, so we dont need to change the values, if you are confused on the the venn diagram simply the part in the middle means how many are in both A and C, meaning that its counting from A and C and you get 30 so you dont have to change the values of A or C
rampage14x13 You are completely wrong. It never said “ONLY algebra” or “ONLY chemistry” students, thus it’s implied that it can include both or either algebra and chemistry students. So it should’ve been 120 students ONLY in algebra, 50 students ONLY in chemistry, and 30 students in BOTH algebra and chemistry. Because if the first question is asking for the probability of algebra students, we are looking at 150 students [out of 500] that include students taking ONLY algebra (120) and students taking algebra and chemistry (30) cause nonetheless, if you’re taking algebra and chemistry, then you’re STILL taking algebra.
@@candykiss024 Yup, I was trying to explain this to the people on her and I got cursed at. You are absolutely right. Gotta be careful with these online sources.
Thank you very much for putting in so much effort. It is clear,precise and easy to understand. Can you send these videos on probabilty? I am willing to pay.
You are given three coins: one has heads on both faces, the second has tails on both faces, and the third has a head on one face and tail on the other. You choose a coin at random and toss it, and it comes up heads. The probability that the other face is tails is
hi, i am a tad confused with the beginning. when you say event B has occurred do you mean the die has been rolled and ? if it is then if event A is either rolling a 1, 3 or 5 it doesn't matter what was rolled before, doesn't it ? thanks for your videos, excellent just this one confused me
At 5:03 why are we dividing 150/500 instead of 180? Why aren't we counting the students that are taking both Algebra and Chem since they would be part of the group that is taking Algebra?
The answer to your first question is wrong. It mentions to find the number of students taking algebra (not ONLY algebra). Thus, we must also include the intersection of algebra and chemistry which would be 180/500.
From my understanding he's not wrong, 150 is the total students taking algebra. The question said "150 students are enrolled in Algebra course" that means all students in algebra, later it mentioned 30 takes both algebra and chemistry stil they're within 150 taking algebra... I know You were confused from the ven diagram he was supposed to write 120 out of that 30 intersection
I think the venn diagram u drew for the school question is wrong. Isn't it supposed to be 120 in the algebra, 50 in chemistry, and 300 on the oustide. Because otherwise the diagram would imply that there are 760 students and not 500?
Probability of P(A) is not 180/500? The same applies to P(C): 110/500. Or A (Algebra) should be 120 + 30 (both A and C)? Im confused. Pls. help. I believe the Venn diagram is wrongly depicted.
It isn't wrong. Think of it this way: X= Students who ONLY take algebra, Y= Students who take BOTH algebra and chemistry and let Z= Students who ONLY take chemistry x+y=150 y+z=80 Since y is given to us as 30, what you get is: x+30=150 Thus after subtracting 30 from both sides: x=120 students that ONLY take algebra And in the end what happens is that 120 (students who only take algebra) +30 (Students who take algebra and chemistry)= 150 TOTAL STUDENTS that take algebra. Although the venn diagram ends up being counter intuitive, the 30 is not additive with the 150, it's just PART of the 150 students. Thus 150/500 is the solution to 1a.
The 30 is included in 150. Only 120 are taking algebra but not taking chemistry. Therefore, total number of students taking algebra is 150 while total number of students taking chemistry is 80
The answers are right but they way the Venn's are drawn is wrong because: Algebra only: 120 students Chemistry only: 50 students Who are taking both: 30 students And the complement of( algebra U chemistry)' : is 300 students (students who are doing neither of algebra or chemistry⚗️)✓✓✓
Okay so what is the probablity when we take A given B is taken a step further like when A and B coexist within and encompass each other, is this to you all impossible I think not as if we go into biology class A and B will coexist and occupy the same space every time a sperm fertilizes an egg in the creation of a new life. does this mean that the probable outcome will contain all of A and B
vann diagram is incorrect. 150 taking algebra, 80 taking chemistry, 30 taking both.. 150 - 30= 120 80 - 30= 50 30 is overlapping in both cases. leaving 120 in A and 50 in B but I see for simplifications he used the 150 30 80 van diagram.
wouldnt u take away the number of students in the chemistry and algebra course from the total number of students and also wouldnt u take away 30 from both chemistry and algebra to show the 30 who take both?
Probability Formula Sheet: bit.ly/3zb22rW
Final Exams and Video Playlists: www.video-tutor.net/
Full-Length Videos & Worksheets: www.patreon.com/MathScienceTutor/collections
Next Video: ruclips.net/video/lWAdPyvm400/видео.html
Good day Sir. Thank you for all you do. Your channel have been of immense help to me various ways, more help than my school lecturers ever were. Thanks so much, may God multiply the blessings in your blessed life on my behalf, Amen.
Please Sir, I need your clarification on question 2 B 12:18 . To me, you are a demi maths god but I beg to disagree with you on it. In my thought, it's either you want to test if we are paying detailed attention to the formula or you got it wrong. From what I have learnt from you and what I know in probability, the solution should be as follows:
P(Br|M) = 70/90 = 7/9
Or
P(Br|M) = P(M and Br)/P(M)
=(70/200)/(90/200) = 70/90 = 7/9
= 0.7778 = 77.78%.
Please correct and clarify me if I am wrong. Please tag me when you do. Thanks
4:52 i swear u draw the venn diagram wrong?
Hey bro in 14:50 seconds why are you doing it differently compared to question C because they are the same questions
Dude, your videos are always so clear and to the point. You're a godsend. Crazy how it doesn't even require any fancy animations or anything, just clear examples and explanations. You are a fantastic teacher!
He is. And knows SO much too
CANNOT THANK YOU ENOGUH!! i hope you finish the series before my exam
Let’s hope this amazing teacher starts teaching English so you can spell ‘enough’
@@somxr_738 I speak 7 language what do you do Tom correct people on the internet pathetic 🤡
Let's not forget to "apply cold water to burned area"
@@somxr_738 BRUH LMFAOOO
@@somxr_738 Bro replying to a comment from.....5yrs ago!!! (I'm doin the same thing lol)
there is a 0% probabillity that I pass math class
😂
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Now you understand the probability
Also you can say "Impossible Event".
no literally i hate statistics
Omg THANK YOU SO MUCH FOR THIS LESSON. I literally sat in my math class for an hour and 40 min straight and I can't understand anything that my teacher is saying. Literally your video is less than 20 min and I understood right away!
I just started year 11 and this video saved me already 💀. I think my survival is going to depend on this channel 🙏
I don't know why I pay tuition fees for my university faculty. If i don't understand anything at the end of the day, i have to come to this RUclips channel and make a solution. And things are very easy to understand. If The Organic Chemistry Tutor had been my university faculty, I probably wouldn't have had to struggle so much. May Allah grant you long life❤️❤️❤️❤️
Learning probability is much better here than in my school/college...I use to ran away from this subject during those days...!Thank you Teacher :)
whenever I don't understand a topic, I come and watch ur videos. they really help me a lot .. u are the best, thank u so much for your videos.
Me spending math online class for 2 periods 😶 vs me spending math class with this guy 😲.
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THANK YOU FOR MAKING MY LIFE EASIER!!! I DON'T UNDERSTAND ANYTHING THAT I LEARNT DURING ONLINE CLASSES 😭
mee tooooo
I watched many of your videos. This is a comment passing by. YOU'RE AMAZING!
I'm very curios to this narrator is he a teacher a professor?
You are amazing! I wish you solved some college probability problems that are in exams. I have an exam soon😢
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You are the greatest person in my life right now, Julio. you never disappoint. Your voice makes it infinite times better!
Thank you very much for your video. Now I clearly understood conditional probability with Venn diagram amd Contingency table. Once again I am so glad and thank you very much
Wow he explained way better than anybody else so far
OMG BETTER THAN MY MATH TEACHER. SHE HAD NO IDEA HOW TO EXPLAIN THE FORMULA AND OUR TEST IS TOMORROW, AND YOU LITERALLY JUST SAVED MY REPORT CARD.. TYSM!!!
I really do not understand why some people dislike this unique content from a really unique teacher !!! 105
One of the best and clearest videos I have seen on this complicated subject.Finally, every calculation fits together. Thank You!!
wait... isn't it A=120 and C=80 since the problem didn't have the word "only". If it is "in an algebra course only" then it is 150
Thanks a lot for comments like this. I thought what I knew was wrong. I had just taught that part to someone and was alarmed when I saw the vid that I might have taught it wrong. Phew.
@@l554446l 😂😂😂 I realized it too
A = 150. A and not B = 120. I wanted to clarify that, but yes, you are right that in a Venn Diagram, we usually split the values by shaded regions, and thus the "A and NOT B" region should have had 120. While A = A and NOT B + A and B = 150.
I was just about to comment thisss
I was so confused for a second
But since 80 students are in chemistry, 30 could also be taking algebra as well? So isn't C = 80-30 = 50???
The venn diagram looks pretty wrong at 5mins
It proved to be an important point of discussion. I was thinking that same. 150 is the number of only Algebra. 180 is the number of Algebra students total.
I love your explanations and ypur channel. But I think in the 1st example you should from 150 substract 30, because these 150 include All students that have Algebra and they also may have Chemistry. and 150-30 it will be 120 students that take Only Algebra. What do you think about it?
The picture isn't drawn quite clear, it'd probably better to put numbers like 120/30/50 i.e. labelling the mutually disjoint parts, I believe it's more intuitive and established notation if we put numbers on Euler-Venn diagrams.
However "the student is taking Algebra" should not be interpreted the exclusive "Only Algebra" so that calculation is correct, whereas if we question the probability of Only Algebra => it will be 120/500 for sure.
When I don't know stuff for math I always look your channel up! Thank you for being better then 99% of the teachers out there 🙏
he's technically the only non indian teacher i feel safe around on the internet lol
Thank you from my whole heart for saving me. You're the GOAT.
Im waiting for statistics and probability videos from you! Thank you so much please make more videos about probability
Can you please do more videos on Probability and Statistics
Thanks a lot I really love your teaching
Your explanation made me love probability
You are a life savior !!!!! My Goodness!!!! Thank you So Much !!!
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This is a fantastic explanation! So can we think of conditional probability as 'changing' the sample space to a new base? For example if we have a 'given it has brown eyes' the new sample space is the column Br and we have to compute any other conditional probability by referring back to this sample space instead of the global sample space.
I agree with you, well said!
One thing that bugs me: If 150 are in algebra, and 30 are in algebra AND chemistry, then wouldnt the probability of selecting a student in algebra be 180/500 = 0.36 ? Then the P(C|A) = 30/180.
its talking about only algebra
Thanks for the help. Now I am confident that I can ace my geometry test coming up
Your such a good teacher. I was confused though about min. 3:15 because you said divided by P(B) however in 6:25 you say P(A). Was just confused about that.
good explanation. however, I have a question what is the difference between: a female with brown eyes and a female given that it has brown eyes? I see they are the same? why have choosen a conditional probablities while I see it is a simple question of a female with brown eyes which should be 100/200 = 0.5 (50%) at 13:40
Thank you so much have now understood conditional probability
YOUR VIDEO IS SO HELPFUL TO ME
no kidding your videos saved my life
Tried Professor Leonard...and am back here..
I think you made an error with the Venn diagram. You should've subtracted the 30 from 150 and 80. Your diagram doesn't seem to represent what the question says.
Exactly my observation.
you're right
Good day Sir. Thank you for all you do. Your channel have been of immense help to me various ways, more help than my school lecturers ever were. Thanks so much, may God multiply the blessings in your blessed life on my behalf, Amen.
Please Sir, I need your clarification on question 2 B 12:18 . To me, you are a demi maths god but I beg to disagree with you on it. In my thought, it's either you want to test if we are paying detailed attention to the formula or you got it wrong. From what I have learnt from you and what I know in probability, the solution should be as follows:
P(Br|M) = 70/90 = 7/9
Or
P(Br|M) = P(M and Br)/P(M)
=(70/200)/(90/200) = 70/90 = 7/9
= 0.7778 = 77.78%.
Please whoever can should correct and clarify me if I am wrong. Please whoever does should tag me when he/she does. Thanks
I don't get something. At (12:16) you compute a union probability, which can be calculated by multiplying the individual probabilities of the events right? Thus, Male bird (90/200) * brown eyes (170/200) = 0.38, not 0.35.
You bloody gem, much thanks.
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Your a legend mate.
I think you did one step wrong, I might be wrong tho. In the question with the students taken Algebra and Chemistry the probability of P(A) should be 180/500 ya? not 150/500. This is because the 30 students also took Algebra as well
Sir, at the 16:26, why aren’t included the 30 students in Algebra?
a really great job
can you please
make a video on conditional expectation?
Quick question about the algebra and chemistry word problem... Since there are 30 students taking both the classes, shouldn't the number of students taking only algebra be 120 students and only chemistry be 50 students?
Arvind Chandrashekar no, the intersection simply means how many students are in both A and C. It was stated in the problem that A has 150 students, C has 80 students, and 30 of those students are in both A and C. The way the venn diagram is a little counter intuitive, but pay attention to what the problem statement says. These things were given to us, so we assume they are true, so we dont need to change the values, if you are confused on the the venn diagram simply the part in the middle means how many are in both A and C, meaning that its counting from A and C and you get 30 so you dont have to change the values of A or C
rampage14x13 You are completely wrong. It never said “ONLY algebra” or “ONLY chemistry” students, thus it’s implied that it can include both or either algebra and chemistry students. So it should’ve been 120 students ONLY in algebra, 50 students ONLY in chemistry, and 30 students in BOTH algebra and chemistry. Because if the first question is asking for the probability of algebra students, we are looking at 150 students [out of 500] that include students taking ONLY algebra (120) and students taking algebra and chemistry (30) cause nonetheless, if you’re taking algebra and chemistry, then you’re STILL taking algebra.
@@candykiss024 Yup, I was trying to explain this to the people on her and I got cursed at. You are absolutely right. Gotta be careful with these online sources.
Don't want to hate but your second question is wrong. Not the answer the Venn diagram
You like for me angels come from. God. Thanks. Teacher. ❤🇪🇹🙏
Thank you very much for putting in so much effort. It is clear,precise and easy to understand. Can you send these videos on probabilty? I am willing to pay.
You are given three coins: one has heads on both faces, the second has tails on both faces, and the third has a head on one face and tail on the other. You choose a coin at random and toss it, and it comes up heads. The probability that the other face is tails is
Amazing teacher
I have really understood. Thank you
hi, i am a tad confused with the beginning. when you say event B has occurred do you mean the die has been rolled and ? if it is then
if event A is either rolling a 1, 3 or 5 it doesn't matter what was rolled before, doesn't it ?
thanks for your videos, excellent just this one confused me
thanks a lot i got it 100 percent
Thank you very much for this
Thank you! You always help
2nd viewer from somalia
Hard work hello
posterior probability is the main purpose of this sort of notin of probability
i love you you helped me so much !
Omg ~ 😲😲what a amazing explanation
At 5:03 why are we dividing 150/500 instead of 180? Why aren't we counting the students that are taking both Algebra and Chem since they would be part of the group that is taking Algebra?
Simple question: rather than referring to A and B as events, isn't it more accurate to call A and B *conditions* of a particular event?
No, since an event is any subset of a sample space.
Thx for this
The answer to your first question is wrong. It mentions to find the number of students taking algebra (not ONLY algebra). Thus, we must also include the intersection of algebra and chemistry which would be 180/500.
From my understanding he's not wrong, 150 is the total students taking algebra. The question said "150 students are enrolled in Algebra course" that means all students in algebra, later it mentioned 30 takes both algebra and chemistry stil they're within 150 taking algebra... I know You were confused from the ven diagram he was supposed to write 120 out of that 30 intersection
Thank you
I think the venn diagram u drew for the school question is wrong. Isn't it supposed to be 120 in the algebra, 50 in chemistry, and 300 on the oustide. Because otherwise the diagram would imply that there are 760 students and not 500?
Probability of P(A) is not 180/500? The same applies to P(C): 110/500. Or A (Algebra) should be 120 + 30 (both A and C)? Im confused. Pls. help. I believe the Venn diagram is wrongly depicted.
Thank you so much
I'm pretty sure the answer to A is 180/500
It isn't wrong.
Think of it this way: X= Students who ONLY take algebra, Y= Students who take BOTH algebra and chemistry and let Z= Students who ONLY take chemistry
x+y=150
y+z=80
Since y is given to us as 30, what you get is:
x+30=150
Thus after subtracting 30 from both sides:
x=120 students that ONLY take algebra
And in the end what happens is that 120 (students who only take algebra) +30 (Students who take algebra and chemistry)= 150 TOTAL STUDENTS that take algebra. Although the venn diagram ends up being counter intuitive, the 30 is not additive with the 150, it's just PART of the 150 students.
Thus 150/500 is the solution to 1a.
The 30 is included in 150. Only 120 are taking algebra but not taking chemistry. Therefore, total number of students taking algebra is 150 while total number of students taking chemistry is 80
@@nellvincervantes3223 The should wrote on the A is 120 right?
Hi can you make a video on possibility diagrams?
Hi. Can we associate a confidence level to conditional probability?
Dang!! You just really do make things simple!! Too bad you professor with his "PH-D can't teach!!!
Exactly
The answers are right but they way the Venn's are drawn is wrong because:
Algebra only: 120 students
Chemistry only: 50 students
Who are taking both: 30 students
And the complement of( algebra U chemistry)' : is 300 students (students who are doing neither of algebra or chemistry⚗️)✓✓✓
This is so good, ty
Okay so what is the probablity when we take A given B is taken a step further like when A and B coexist within and encompass each other, is this to you all impossible I think not as if we go into biology class A and B will coexist and occupy the same space every time a sperm fertilizes an egg in the creation of a new life. does this mean that the probable outcome will contain all of A and B
Why didn't you include the students taking Algebra AND Chemistry for part A in the first question? I thought it would be 180/500 not 150/500
Can you please explain the a creature with blue eye given that is a female isn’t it supppose to be 30/30 I’m confused
The second question’s Venn diagram doesn’t make sense.
thank u soo so so much
I'm interested in this black writing tool... which app/website is this?
6:07 i still don't understand why the denominator is only 150. why would he only count the students taking algebra as the maximum population?
I like it sir
As always, brlliant!
What 's the difference between a male with brown eyes and a female given it has brown eyes ?
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My teacher said that the equation is "P(A and B)/P(A)", not "P(A and B)/P(B)". Does it change anything?
Your teacher must've written P(B and A)/P(A) then it's correct
Thanks dude!
06:11 Why dont you calculate 30/180 ?
Contingency tables are fun
vann diagram is incorrect.
150 taking algebra, 80 taking chemistry, 30 taking both..
150 - 30= 120
80 - 30= 50
30 is overlapping in both cases. leaving 120 in A and 50 in B
but I see for simplifications he used the 150 30 80 van diagram.
Also, 500 must be subtracted by the total of the students enrolled in Algebra and Chemistry.
I don't agree though with 150 30 80 to make the explanation easier. Because it would create misconceptions to the learners.
thank youu
thanks
wouldnt u take away the number of students in the chemistry and algebra course from the total number of students and also wouldnt u take away 30 from both chemistry and algebra to show the 30 who take both?
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