The Law of Total Probability | Probability Theory, Total Probability Rule
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- Опубликовано: 25 июл 2024
- What is the law of total probability? Also sometimes called the total probability rule, we go over this tremendously useful law in today’s full video math lesson! Imagine we have a sample space that can be partitioned into three events B1, B2, and B3. And say we have another event in this sample space called A. Then, whenever A occurs, it must be true that B1 also occurs, or B2 also occurs, or B3 also occurs, by definition of a partition. It also follows from the definition of partition that B1, B2, and B3 are disjoint, no more than one of them can occur at the same time.
Therefore, we can find the probability of A by adding the probability that A and B1 occur, the probability that A and B2 occur, and the probability that A and B3 occur.
That is, P( A ) = P( A intersect B1 ) + P( A intersect B2 ) + P( A intersect B3 ). This is the law of total probability for a partition into three sets, but it is also true for any partition into a finite number of sets, or a countably infinite number of sets!
So if we know the probabilities of those intersections, we’re good to go, but often times we do not. Instead we use the formula for the probability of an intersection that uses conditional probabilities! Recall the definition of conditional probability, P( A | B ) = P( A intersect B ) / P( B ), which gives us the equation P( A intersect B ) = P( A | B )*P( B ). So if we know the probability of A given certain conditions, we can use this equation and apply the law of total probability.
The law we previously wrote as: P( A ) = P( A intersect B1 ) + P( A intersect B2 ) + P( A intersect B3 ) then becomes:
P( A ) = P( A | B1 )*P( B1 ) + P( A | B2 )*P( B2 ) + P( A | B3 )*P( B3 ) .
SOLUTION TO PRACTICE EXERCISE:
We are given that 40% of students are male and 60% are female. Sixty percent of males are taller than 6 feet, meaning 40% of the males are shorter than 6 feet. Ten percent of the females are taller than 6 feet, which implies 90% of the females are shorter than 6 feet. We want to find the percent of the class that is shorter than 6 feet.
The sample space is partitioned into males and females. Let E be the event that a student is shorter than 6 feet, M be the event that a student is a male, and F be the event that a student is female. Then, from the law of total probability, we have that P( E ) = P( M intersect E ) + P( F intersect E ). In this case we don’t know the probabilities of these intersections but we do know some conditional probabilities. Rewriting the equation using conditional probabilities, we have that P( E ) = P( E | M )*P( M ) + P( E | F )*P( F ). Thus, P( E ) = 40%*40% + 90%*60% = 16%+ 54% = 70%.
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If you are preparing for Probability Theory or in the midst of learning Probability Theory, you might be interested in the textbook I used when I learned Probability Theory. It is "A First Course in Probability Theory" by Sheldon Ross. Check out the book and see if it suits your needs! You can purchase the textbook using the affiliate link below which costs you nothing extra and helps support Wrath of Math!
PURCHASE THE BOOK: amzn.to/2Z8MrB7
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I hope you find this video helpful, and be sure to ask any questions down in the comments!
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I've watched countless maths tutorials on RUclips, and this is undoubtbly the clearest explaination of a concept that i've encountered. Got 70%! Thank you!
That means a lot, thank you Yen! So glad it was clear - let me know if you ever have any video requests! If you're in the mood for some less clear explanations but with more rhymes, check out my math songs! ruclips.net/p/PLztBpqftvzxW7a66b0dJPgknWsfbFQP-c
Can't believe it only took you 10 minutes! Better than textbook any day
I routinely sacrifice brevity for clarity so I am glad you found this lesson clear and brief! Thanks for watching!
This is one of the best teachers in the world. God bless you.
I've been hopping around trying different explanation videos for statistics and this is by far the simplest and most engaging. I've been struggling with distance learning and feeling like I just wouldn't be able to "get" statistics. Thank you so much! I love your animation, color coding, and the simple practice problems. Thank youuuuuuu!
Thank you so much for the kind words, Stephanie! I'm very glad you found the lesson helpful and wish you the best in your remote learning! Keep working hard, and don't be afraid of your textbook if you have one! There are also some good online textbooks for statistics you may find valuable, check out OpenStax stats textbooks.
I took a mathematical statistics class at UCF ten years ago and I never understood The Law of Total Probability until now. This came up in a Bayesian statistics class today that I'm currently taking for graduate school preparation, so I decided to youtube the topic because it still didn't make much since to me. I found your video and it totally makes sense to me now! Where were you ten years ago when I needed you Wrath of Math? Thanks for the time and effort that you put into this video explaining The Law of Total Probability.
Thanks so much for watching Rodney! One of the nicest comments I've received, so glad it helped! Ten years ago I was probably just beginning to study algebra in middle school, but I'm here to stay now! Best of luck in your preparation for graduate school!
Your videos are a hidden gem! 🙌🙌🙌
Simply incredible! U made it so easy to understand. I also loved the exercises that you did. Can't wait to see more from this channel :)
So glad it helped! Thanks a lot for watching; I release new lessons every 48 hours :) Let me know if you ever have any requests!
Great,very educative.God bless you.
Thank u for explaining this concept in such a fun and easy way.
Thanks, very helpful and easy to understand
The answer for the question in the last video as the following:
Firstly, in question it is asking us “What Is The Probability of Shorter Than 6 Feet” and in the same time it given us 4 things as the following:
1. P(male) = 0.4
2. P(female) = 0.6
3. P(taller than 6 feet | male) = 0.6
4. P( taller than 6 feet | female) = 0.1
# From 3 and 4 we have to know directly that :
P(shorter than 6 feet | male) = 100/100 - 60/100 = 0.4
P( shorter than 6 feet | female) = 100/100 - 10/100 = 0.9
*Let “male” denotes by “m”.
*Let “female” denotes by “f”.
*Let “Shorter than 6 feet” denotes by “S”.
*Let “Intersection” denotes by “&”.
After we extracted the givens from the question, Now we can apply “ The Total Probability Theorem “. to get “The Probability of Shorter Than 6 Feet”. As the Following:
P(s) = P(s&m) + P(s&f)
*From The Conditional Probability Rule : P(A|B) = P(A&B) / P(B)
P(A&B) = P(A|B) . P(B)
So , P(s) = P(s|m). P(m) + P(s|f). P(f)
P(s) = (0.4)(0.4) + (0.9)(0.6) = 0.7
So finally “The Probability of Shorter Then 6 Feet” is {0.7}
I hope you all did understand my answer, Regards 2022
wow!! I love your work. Please I'm currently taking a Bsc course in Stat 111 and I have questions to ask here later thanks.
Well done man
Thanks man!
thanks
Fail
Thank you for your help!
That was an impeccable explanation
Thank you. This was helpful
Thank you so much for this, how you simplify things and explain fully is simply amazing, be blessed!
So glad it helped! Thanks a lot for watching!
thank u sir. your explanation is crystal clear. please do some examples problems on total probability
Gr8 video. Haven't seen any so simple and effective in explaining the law of total probability.
Thank you!
interesting video Wrath of Math. I broke that thumbs up on your video. Continue to keep up the great work.
Thank you so much , it was so helpful. I wasted my 3 days in searching of this topic and finally I found a good one Thank you 🙏🏻
You're very welcome, I'm so glad it helped! Thanks for watching!
Clear thank you
I have got 70 percent as the answer and thanks for such a nice explanation ...Totally understood the Total probability law..
Got the same answer! Cheers
thanks a lot love you !
u make my life easy 😉 thanks with respect 🙂
Easy explanation, i easily grasped the concept, thank you for uploading for such an easy explanation video 😊, have a nice day 👍😊
Thank You
I have loved the way you have explained.Bravo!
Thank you!
Very helpful! Thank you
Glad it was helpful!
Amazing video!! This channel is under rated, will share the video with my friends
Thank you!
Thanks for this, its so helpful to me!
Glad to hear it, thanks for watching!
Thanks, your videos are helpful :)
You’re welcome and thanks for watching! I’m glad you’re finding them helpful :)
much love 👍, you are indeed a "wrath of math", my learning place thanks 🙏.
So glad to help!
You made it easier to understand. Thank you
Glad to help!
lifesaver!
Very easy to understand. thanks
Glad it helped, you're welcome and thanks a lot for watching!
Thank you so much! Its Soo clear now
Glad it helped, thanks for watching!
Thank you so much for this
Glad to help!
I was really finding it difficult to understand the concept, lest apply the same. The explanation given here is really easy to understand and apply too. Thank you Sir. Got 70 as an answer. Was new to this channel and now I would recommend to all...
Thanks for watching! This can certainly be a tricky topic, I am glad the lesson helped! Let me know if you ever have any questions, I don't have a huge probability playlist yet, but I'm always happy to dip back into the topic if I can to cover something.
Best explanation I've found
Glad it was helpful!
wonderful explanation
Thank you!
thanks
Thanks so much for this. i had to take a student this topic but i had no idea but now i am good.
My pleasure, thanks for watching!
Well done !
Thank you!
Take,
S - students shorter than 6 feet
M - male students
F - female students
P(S) = P(S|M).P(M) + P(S|F).P(F)
=40%*40% + 60%*90%
=70% or 0.7
But solution given in the description has some error which is giving wrong answer (78%).
BTW thanks for the video.
You’re exactly right! Thanks a lot for watching and for pointing out that error, it has been corrected! You’re very welcome, I appreciate your support!
Thanks so much, in fact when i was thought during lectures i saw and understood nothing but thankfully u hv help me ...... however i had 90% as thr percentage of those less that 6 feet am I correct
Make video of difference between Bayes theorem and total probability theorem
thanks mate
Glad it helped!
ty! i dont get why they have to make it so hard to understand in school when it can be simplified and taught like this :)
So glad it helped! You're welcome and thanks for watching!
you are awesome
thanks man
My pleasure, thanks for watching!
The best!!!!!!!!!!
Thank you!
70%. Thanx this is superb
Glad to help!
Sir you are just too good in this.🙌
Thank you so much 😀
Can you made video on regression line topic please ?
I got it right
I've my maths exam day after tomorrow..
Awesome explanation, Thankyou Sir♥️
Thank you! I hope it went well!
@@WrathofMath it went well.. ✨💖
Great
Thank you!
Precise lectures❤📈🔥
Thank you!
@@WrathofMath 😊
Brilliant
Thank you! Let me know if you ever have any questions!
@@WrathofMath I'd love for you to explain the moment generating functions in probability distribution 😬
obrigado❤👏
Meu prazer, obrigado por assistir!
70% of the class is smaller than 6 feet
Just wow
Thanks for watching!
That was super clear
BTW English is my second language, nevertheless I understood everything
So glad to hear it! Thanks for watching!
How did you get 40% and 90% for the solution to the practice problem?
Amazing^_^
Thank you!
is this the total probably?
It is good that here you only connect the gender with height. In my probability class in Hong Kong, my professor loves to connect gender with major (strong stereotype..
8:24
Thanks. The answer is 70 percent of class.
( 0.4×0.4)+(0.6×0.9)=0.7=70%.Therefore 70% are shorter than 6 feet
. The deck of sixteen cards shown in #2 is thoroughly shuff led. Three cards are drawn from t he
top of the deck, one at a time. What is the probabilit y t he third card is an ace?
What is #2?
@@WrathofMath divided in 2 parts
The question can be answered intuitively too.
Say the total is 100 people. Then 60% of it female; that is 60 girls. Then there are 40 boys.
Out of 60 girls, 10% are taller. 10% of girls means just 6. Therefore 54 girls are shorter. In the same way, 60% of 40 boys are taller; as a number it’s 24 boys. So, 16 boys are shorter. Thus, 54+16 = 70 people are shorter, out of 100 people. As a percentage it’s 70%.
🙇🙇
for the last example , I got 0.6 is it correct?
No it is %70
i got 78%
70%
Thanks for the video! I have a question for the example. Can we say that P(R) = P(R∩A)+P(R∩B)+P(R∩C) instead of P(A∩R)+P(B∩R)+P(C∩R)?
Yes we can, the intersection operation is commutative - so the order of the sets does not matter!
Thank you so much for the clarification! Do you have a video explaining conditional independence?
70
0.6x0.4 + 0.1x0.6 = 0.3 Means 30%
70% are shorter than 6 feet
Your video seemed to make sense to me, but it didn't help me to understand what I was trying to understand: ruclips.net/video/0lKHFJpaZvE/видео.html
With his "plain vanilla application of the law of total probability", Cyrill introduces a new variable out of nowhere (x_{t-1}) and integrates over it, and I have no idea why or how this connects with the law of total probability, Also, he has multiple givens, and I'm not sure how to work with those. Any recommendations to help me understand that? Thanks!
I got 0.7
Ball bag 1, Ball bag 2, Ball bag 3
I can solve the problem without using probability. you can just assume there are 100 students in total and come with 70% are shorter than 6ft
i got 0.7
Exactly! Good work and thanks for watching, let me know if you have any lesson requests!
I had 0.868
I had 0.7
🤷🏾♂️🇿🇲❤️✌️
Let Probability of Being Male be P(M)=0.4
probability of Being Female be P(F)=0.6
Probability of Being short and male P(S|M)=0.4
Probability of being short and female P(S|F)=0.9
Therefore P(S)= P(S|M)P(M) + P(S|F)P(F)
P(S)=(0.4x0.4) + (0.9x0.6)
P(S)=0.7
70% of the class is shorter than 6 feet