Multiplication Rule of Probability | Probability Theory, Intersection of Two Events
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- Опубликовано: 21 авг 2024
- What is the multiplication rule of probability? How do we find the probability of the intersection of two events? That’s what we’ll go over in today’s probability video lesson! The multiplication rule can also be generalized to apply to the intersection of more than two events, but we’ll talk about that in another lesson!
If two events, A and B, are independent, then P(A and B) = P(A)*P(B). This is fairly intuitive when we look at some examples like the flipping of two coins for example. What is the probability that the first coin lands heads up? That would be 1/2. What is the probability that the second coin lands heads up? That is also 1/2. The events are independent. Flipping heads on the first coin does not change the probability of the second coin landing heads up. What is the probability of both coins landing heads up? That would be (1/2)*(1/2) = 1/4.
However, things get more complicated when we deal with dependent events. This is because if an event A has occurred, and it is not independent from B, then the probability of B is affected by the occurrence of A. Check out the full video lesson for an explanation of this situation, and how the multiplication rule applies! The multiplication rule in this situation uses what is called a conditional probability.
Check out my lesson on conditional probability: • Intro to Conditional P...
SOLUTION TO PRACTICE PROBLEM:
There are two ways that one ball from the jar can be green and one can be red. Either the first is red and the second green, or the first is green and the second red.
A: First ball is red
B: Second ball is green
Then P(A) = 7/17 (number of red balls over total number of balls) and P(B | A) = 10/16 (number of green balls over new total number of balls). Thus, P(A and B) = (7/17)*(10/16) = 70/272. We can calculate the probability of the first ball being green and the second being red in a very similar way, and it is exactly the same probability. Try verifying this fact yourself. Thus, the probability that one ball is red and one is green is 70/272 + 70/272. = 140/272. To check our answer, we could calculate the probabilities of the rest of the possible outcomes of drawing two balls (the other possibilities are that they are both red or that they are both green) and then add those probabilities to 140/272. The total sum should be 1. Try checking your answer this way!
If you are preparing for Probability Theory or in the midst of learning Probability Theory, you might be interested in the textbook I used in my Probability Theory course, called "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/31mXEjr
I hope you find this video helpful, and be sure to ask any questions down in the comments!
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Thank you for your help, I attended my lecture at the university, and also I have watched many videos and I didn't understand, but you explained this rule in a very simple way and was understandable, thank you again!
Thanks for watching, I'm glad it helped!
Lets A be probability of Red to occur
P(A) = 7/17
Let B|A be the probability of Green to occur when red has already occured.
P(B|A) = 10/16
SO, P(A n B) = 7/17 *10/16
Ans =25.7%
Is this correct Sir?
I think you can either have the 1st ball being red and 2nd ball being green OR the 1st ball being green and 2nd ball being red. So given this the probability I think should be 7/17 * 10/16 + 10/17* 7/16
Short,brief and precise video keep it up man
God bless you. Thank you for dummy proof explanations.
Thank you for watching! I try to just make them highly detailed and very clear haha, I consider it for people with fine taste in math videos rather than dummies 😂
Would anything change if the cards in the considered example were drawn simultaneously? How would it change the probability calculations? Thank you!
You are appreciated!! Thank you!!
The probability that one ball is red and one ball is green should be around 51.33%.
Hey can u provide me solution of last example that u have done in this video
What's the final answer sir
Given balls:-
Red = 7 and Green = 10
Total balls = 17
P(A) = 7/17
P(B) = 6/16
P(A and B) = 7/17 * 6/16 = .003
Is this correct..?
Thanks for watching, Ricky. I'd like to help with your question, but the answer depends on what A and B are exactly. Is A the event that the first ball is red and B the event that the second ball is red?
@@WrathofMath b is the event where the second ball is red
Nice work
Thanks a lot!
Excellent ❤
Thank you!
Excellent!!
Thank you!
@@WrathofMath Can you upload video on advanced topics like Poisson, Normal etc distributions?
Thx a lot !!!!! ❤️❤️❤️❤️
No problem, thanks for watching!
I got 51.47% as the final answer
im sorry to say but you, along with the two other idiots who liked this comment; were infact incorrect 🤓🤓
I kind of know it and able to solve it..yet I don't understand it fully... am I dumb ?😅😅
Awesome.video
Thanks for watching!
Please I need the answer of last question tomorrow I have exam
7/17×10/16(because of without replacement)
👍🏾
Thanks for watching!
Legend❤🫡📈🔥