Venn diagrams and adding probabilities
Addition Rule for Probability Venn diagrams and the addition rule for probability
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- Let's say I have a bag and in that bag I am going to put some green cubes
- and in particular I am going to put 8 green cubes.
- I am also going to put some spheres in that bag.
- Let's say I am going to put 9 spheres and these are the green spheres
- I am also going to put yellow cubes in that bag
- Some yellow cubes. I am going to put 5 of those.
- And I am also going to put some yellow spheres in this bag.
- Yellow spheres. Let's say I will put in 7 of those.
- I must take them all in this bag. And I am going to shake this bag and
- then I am going to pour them out and I am going to look
- at the first object that falls out of that bag.
- What I want to think about in this video is:
- What are the probabilities of getting different types of objects?
- So for example what is the probability,
- what is the probability of getting a cube?
- A cube of any, of any color.
- What is the probability of getting a cube?
- Well to think about that, we should think about what -
- Or this is one way to think about it. -
- What are all of the equally likely possibilities that might pop out of the bag?
- Well we have 8 plus 9 is 17, 17 plus 5 is 22.
- 22 + 7 is 29. So we have 29 objects. There are 29 objects in the bag.
- Did I do that right? This is 14, yep, 29 objects.
- So let's draw all of the possible objects.
- And I will do this in, I will present this in this big area.
- I will represent it as this big area right over here.
- So these are all the possible objects.
- There are 29 possible objects.
- So there is 29 equal possibilities, for the outcome
- of my experiment of seeing what pops out of that bag.
- Assuming that it is equally likely for a cube or a sphere to pop out of that bag first.
- And how many of them meet our constrain of being a cube?
- Well I have 8 green cubes and I have 5 yellow cubes.
- So there are a total of 13 cubes.
- So let me draw that set of cubes.
- So there is 13 cubes.
- Let's draw it like this.
- There are 13, 13 cubes.
- This right here is the set of cubes.
- And I am not drawing it exact, I am approximating.
- This represents the set of all the cubes.
- So the probability of getting a cube,
- is the number of events that meet our criteria,
- so there is 13 possible cubes that have an equal likely chance of popping out.
- Over all of the possible equally likely events, which are 29.
- That includes the cubes and the spheres.
- Now let's ask a different question.
- What is the probability of getting a yellow?
- A yellow object. Either a cube or a sphere.
- So once again, how many things meet our conditions here?
- Well we have 5 plus 7; there is 12 yellows objects in the bag.
- So we have 29 equally likely possibilities.
- I will do that in the same color.
- We have 29 equally likely possibilities, and of those,
- 12 meet our criteria. So let's, so let me draw 12 right over here.
- I will do my best attempt. So that it looks something like...
- A set of yellow objects, there are 12.
- There are 12 objects that are yellow.
- So the 12 that meet our conditions are 12 over all the possibilities, 29.
- So probability of getting a cube is 13 over 29.
- The probability of getting a yellow 12 29ths.
- Now let's have something a little bit more interesting.
- What is the probability...
- What is the probability of getting a yellow cube?
- So I will put in yellow. So we care about the color now.
- So this thing is yellow.
- What is the probability of getting a, as my son would say lellow,
- What is the probability of getting a yellow cube?
- Well, there is 29 equally likely possibilities.
- There is 29 equally likely possibilities.
- And of those 29 equally likely possibilities, 5 of those are yellow cubes.
- Or lellow cubes.
- 5 of them. So the probability is 5 29ths.
- And where would we see that on this Venn diagram that I have drawn?
- And this Venn diagram, this is just a way to visualize the different probabilities.
- And they become interesting when you start thinking about where sets overlap.
- Or even where they don't overlap.
- So here we are thinking about things that are members of the set yellow,
- So there in this set, and they are cubes.
- So this area right over here. That is the overlap of these two sets.
- So this area right over here.
- This represents things that are both yellow and a cube.
- Because they are inside both circles.
- So this right over here. Let me write it over here.
- So there is 5 objects that are both yellow and,
- yellow and cubes.
- Now let's ask, and this is probably the most interesting thing to ask,
- What is the probability,
- What is the probability of getting something that is yellow or...
- that is yellow or a cube?
- A cube of any color.
- A cube of any color.
- The probability of getting something that is yellow or a cube of any color.
- Well we still know that de denominator is going to be 29,
- these are all of the equally likely possibilities that might jump out of the bag.
- But what are the possibilities that meet our conditions?
- Well, one way to think about it,
- one way to think about it, is:
- Well, probably, there is 12 things that would meet the yellow condition.
- So that would be this entire circle right over here,
- 12 things that meet the yellow condition.
- So this right over here is 12.
- This is the number of yellow. That is 12.
- And then to that, we can't just add the number of cubes.
- Because if add the number of cubes,
- we have already counted these 5.
- With these 5 are counted as part of these 12.
- One way to think about it is,
- There are 7 yellow objects that are not cubes,
- those are the spheres.
- There are 5 yellow objects that are cubes.
- And then, there are 8 cubes that are not yellow.
- That is one way to think about it.
- So when we counted these 12, the number of yellow,
- we counted all of this.
- So we can't just add the number of cubes to it,
- because then we would count this middle part again.
- So then we have to essentially count cubes,
- the number of cubes, which is 13,
- 13, the number of cubes. Number of cubes.
- Number of cubes.
- I will have to subtract out, this middle section right over here.
- Let me do this.
- So, subtract out the middle section right over here.
- So minus 5.
- So this is the number, the number of yellow cubes.
- It feels weird to write the word yellow in green.
- The number of yellow cubes.
- Or another way to think about it,
- Or we can just do this math right here.
- 12 plus 13 minus 5 is what? It is... It is 20.
- Did I do this right? 12 minus.... Yep it is 20.
- So that is one way, you just get this is equal to 20 over 29.
- But the more interesting thing than, even than the answer of the probability of getting that,
- is expressing this in terms of the other probabilities
- that we figured out earlier in the video.
- So let's think about this a little bit.
- We can rewrite this fraction right over here,
- We can rewrite this as: 12/29, plus 13/29, minus 5/29.
- And this was the number of yellow over the total possibilities.
- So this right over here was the probability of getting a yellow.
- And this right over here was the number of cubes over the total possibilities.
- So this is plus the probability of getting, of getting a cube.
- The probability of getting a cube.
- And this right over here was the number of yellow cubes over the total possibilities.
- So this right over here was
- minus the probability of yellow and, and a cube.
- And I can write it that way.
- The probability of yellow, alright, yellow in yellow,
- yellow and, yellow and getting a cube.
- Yellow and getting a cube.
- So, and what we have just here,
- and you can play with the numbers,
- the numbers I just used as an example right here.
- To make things a little more concrete.
- But you can see this is a generalisable thing.
- If we have the probability of one condition or a number of conditions.
- Let me rewrite it.
- The probability, and I will just write a little bit more generally here.
- This gives us an interesting idea.
- The probability of getting one condition,
- of an object being a member of set A, or, or a member of set B,
- is equal to the probability that it is a member of set A,
- plus the probability that it is a member of set B,
- minus the probability that it is a member of both.
- minus the probability that it is a member of both.
- This is a really useful result,
- and I think it is sometimes called like the addition rule of probability.
- But I want to show you it is a completely common sense thing.
- The reason why you can't just add these two probabilities,
- is because they might have some overlap,
- there is a probability of getting both.
- And if you just added both of these,
- you would be double counting that overlap.
- Which we have already seen earlier in this video.
- So you have to subtract one version of the overlap out,
- so you are not double counting it.
- And I will throw one other idea.
- Sometimes you have possibilities that have no overlap.
- So let's say this is a set of all possibilities.
- This is the set of all possibilities.
- And let's say this is the set that meets condition A.
- And let's say this is the set that meets condition A.
- And let's, this is, let me do this in a different color.
- And let's say this is the set that meets condition B.
- So in this situation there is no overlap.
- There is no way; nothing is a member of both set A and B.
- So in this situation, the probability of A and B is 0.
- There is no overlap.
- And these type of conditions, or these two events,
- are called mutually exclusive.
- Mutually Exclusive.
- So if events are mutually exclusive that means
- that they both cannot happen at the same time.
- It is... There is no event that meets both of these conditions.
- And if things are mutually exclusive,
- then you can't say the probability of A or B is a probability of A plus B.
- Because this thing is zero.
- But if things are not mutually exclusive,
- you would have to subtract out the overlap.
- And the easiest, probably the best to think about it,
- is to just always realize that you have to subtract out the overlap,
- and obviously if something is mutually exclusive,
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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