Algebra II
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California Standards Test: Algebra II
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California Standards Test: Algebra II (Graphing Inequalities
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CA Standards: Algebra II (Algebraic Division/Multiplication)
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CA Standards: Algebra II
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Algebra II: Simplifying Polynomials
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Algebra II: Imaginary and Complex Numbers
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Algebra II: Complex numbers and conjugates
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Algebra II: Quadratics and Shifts
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Examples: Graphing and interpreting quadratics
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Algebra ||: Conic Sections
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Algebra II: Circles and Logarithms
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Algebra II: Logarithms Exponential Growth
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Algebra II: Logarithms and more
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Algebra II: Functions, Combinatorics
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Algebra II: binomial Expansion and Combinatorics
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Algebra II: Binomial Expansions, Geometric Series Sum
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Algebra II: Functions and Probability
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Algebra II: Probability and Statistics
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Algebra II: Mean and Standard Deviation
Algebra II: Probability and Statistics 76-79, probability, mean, standard deviation
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- We're on problem 76.
- It says one bag contains 2 green
- marbles and 4 white marbles.
- So that's one bag, and we have 2 green and 4 white.
- And a second bag contains 3 green
- marbles and 1 white marble.
- So here's a second bag.
- It has 3 green and 1 white marble.
- If Trent randomly draws 1 marble from each bag, what is
- the probability that they're both green?
- What's the probability that he draws a green marble
- from this first bag?
- Let's see, there's 2 green marbles.
- Out of a total how many marbles?
- 2 plus 4 marbles.
- There's a total of 6 marbles in here.
- So there's a 2/6 or 1/3 chance that he draws a green marble
- out of this bag.
- And then when he goes to this bag, let's say, OK how many
- green marbles are there?
- Well there are 3 green marbles out of a total
- possibility of 4 marbles.
- So there's a 3/4 chance that he picks a green
- marble out of this bag.
- So the chance that he picks a green marble out of both bags
- is just the 2/6 times the 3/4.
- That's 1/3 times 3/4.
- Which is equal to a 1/4 probability.
- Next question.
- I've decided that it's fast when I pre- cut
- and paste the questions.
- It took me, like, 18 videos to think of that innovation.
- Anyway.
- A box contains 7 red marbles, no.
- 7 large red marbles, 5-- let me see, we have a box-- I have
- 7 large red marbles, 5 large yellow marbles, 3 small red
- marbles, and 5 small yellow marbles.
- If a marble is drawn at random, what is the
- probability that it is yellow?
- So probability that it is yellow, given that it is one
- of the large marbles.
- OK, so this might seem difficult.
- Conditional probability and all of that.
- But all you have to think about is, OK, we've picked a
- marble out of the box and it's large.
- So if I have a large marble in my hand, what is the
- probability that I picked a yellow large marble?
- So the easiest way to think about it is just think about
- the universe of large marbles.
- We picked 1 of these 12 marbles here.
- We picked 1 of the large ones.
- So we definitely picked 1 of these 12 marbles.
- And so what is the probability that I picked a
- yellow large marble?
- So there's a total of 5 yellow large marbles.
- That's what they want to know.
- Given it was one of large, probability of the yellow.
- So there's 5 yellow large marbles.
- That's these right here.
- And how many total large marbles might I have picked?
- Well 5 plus 7 is 12.
- So there's a 5/12 chance that I picked a yellow marble,
- given that I picked 1 of the large marbles.
- Next question.
- There's a trash truck outside making a lot of noise.
- Hope you don't hear it.
- Anyway, this has a big diagram.
- Let's see.
- Jamie will definitely try out-- this is problem 78-- let
- me switch to a dark color because I'm going to be
- writing on white.
- Jamie will definitely try out for either
- basketball or baseball.
- Basketball or baseball.
- But not both.
- The probability that Jamie will try out for baseball and
- try out for catcher is 42%.
- So this probability is 42%.
- Baseball and catcher.
- What is the probability that Jamie will try out for
- basketball?
- OK, so what have they provided us?
- There's some probability that Jamie tries out for
- basketball, some probability that Jamie
- tries out for baseball.
- Let's just assign some, let's say that this is x.
- Because that's what we want to figure out.
- And if that probability is x, then the probability of trying
- out for baseball is just going to be 1 minus x.
- And if that's 60%, it's going to be 40%.
- I don't know what it is yet.
- But then they say, if Jamie is trying out for basketball,
- then there's an 80% chance that he tries out for guard
- and a 20% chance for forward.
- But, if you said what's the probability that he does
- basketball and forward, it would be x times 20%.
- Maybe there's a 50% chance he does basketball and then 20%
- of those times he would be a forward, so it
- would be a 10% chance.
- So we do the same logic here.
- If he tries out for baseball, there's a 70% chance he'll try
- out for catcher.
- And they already told us that the combined probability, the
- probability that he'll try out for baseball and, given that
- try out for catcher, that that is 42%.
- So we can write that 1 minus x, that's this probability,
- times 70%, times 0.7, is equal to 42%, which is 0.42.
- And then you get 1 minus x is equal to what?
- What's 0.42 divided by 7?
- Let's see, it's 42 divided by 7 would be 6.
- 0.7.
- It would be 60%.
- And you can do that division.
- And it's always good to do a reality check and say, OK, 70%
- of something is 42%.
- OK, 70% of 60%, yeah that makes sense
- that it would be 42%.
- Just in case you don't make a decimal mistake.
- So 1 minus x is 60%.
- This is a 60% probability right here
- going down this chain.
- And so now maybe you can see it a little more concretely,
- we said, OK, whatever this is times the probability that he
- tries out for catcher, given that he's trying out for the
- baseball team, that's going to be equal to 42%.
- And we've figured this out.
- Now if this is 60%, 1 minus x, then x is-- if this is 60% x
- is 1 minus that.
- So x is 40%.
- And we could have actually solved for that.
- We could have actually said, 1 minus x is equal to 60%.
- And then you could have said, minus x is equal to minus 40%.
- Subtract 1 from both sides, x is equal to 40%.
- Anyway that's the question they asked us.
- And what choice was that?
- What choice?
- Oh, it doesn't matter.
- That was choice A.
- Two problems left.
- I'm going to be nostalgic for these California standards,
- Algebra II exams. All right.
- Let me copy and paste this one in.
- All right.
- A small business owner must hire seasonal workers as the
- need arises.
- The following list shows the number of employees hired
- monthly for a five-month period.
- If the mean of these data is approximately 7, what is the
- population standard deviation for these data?
- Round the answer to the nearest decimal.
- OK.
- Standard deviation has all sorts of definitions.
- Well there is one definition for it, but I just always
- think about it is, it's the average
- distance from the average.
- Or it's the average distance from the mean.
- And if you wanted a visual kind of representation of it,
- let's say that I were to draw-- this is some type of
- y-axis there-- and on the x-axis I were to draw each of
- the data points.
- So this is data point 1, this is data point 2.
- You wouldn't have to do this on the problem, but I really
- want to give you an intuition for what the standard
- deviation is so that you don't forget it when
- you're 32 years old.
- All right.
- So the data points go all the way from 4 to 13.
- So let's see, data point one is 4.
- Let's say that this is 4.
- I just want to give you.
- So that's 4.
- Data point two is 13.
- So maybe 13 will be up here.
- Data point two is 13.
- Data point three is 5.
- That's right around there.
- Data point four is 6.
- Maybe it's right there.
- And data point five is 9.
- So maybe nine is-- I don't know I'm just eyeballing it--
- maybe 9 is right there.
- And they're telling us that if we average these numbers, our
- average is 7.
- So 7 is our average.
- So this is 7.
- So the average is 7.
- And they didn't have to tell you that.
- They're just trying to save you some time.
- You could have added all of these numbers up, and you said
- OK, I got 7.
- I'm guessing that we get some decimals or something.
- That's why they said approximately 7.
- But anyway, that's the average.
- So the standard deviation is just the average distance from
- the average, or the average distance from the mean.
- So you say is, and you might know it as the square root of
- variance and all that.
- And the variance is just the square of the
- distance from the mean.
- And they just really square it so that it stays positive.
- But if you just say 4 and 7, what's the difference, what's
- the distance between 4 and 7?
- Well I'll draw it with my marker.
- The distance between 4 and 7 is 3.
- Remember we just want to figure out the distance.
- We don't want to know which one's bigger.
- 4 is 3 away from 7.
- That's why sometimes people will say, oh OK, the variance
- is equal to 4 minus 7, squared.
- And then that'll give you 9.
- And then they will say that the standard deviation is
- equal to the square root of the variance.
- Which is equal to 3.
- But you could have just said that the standard deviation is
- equal to the absolute value of 4 minus 7.
- To me, this is more intuitive because we're just saying, OK,
- how far away is it from 7?
- I don't care if it's above or below.
- I just want to know the distance from it.
- And then I just want to average each of those.
- Oh I'm sorry.
- This isn't the standard deviation itself.
- This is each of the terms and then we would average them.
- Anyway, let me erase this because I think
- I'll need that space.
- The standard deviation, you figured this out, you take the
- sum of these and then you average them.
- I'll show it to you in a second.
- Let me erase this.
- I think the proper notation would have been, you could
- have written it as the standard deviation is equal to
- the sum of the differences of the data points.
- You know, there's n data points.
- The n data points.
- And that we want to take the absolute value between each of
- the data points.
- Data point i minus the mean.
- They write mu for the mean, or something.
- It gets complicated.
- So that's the sum of all of them and then you want to
- divide that by n.
- So essentially you're taking the average
- distance from the mean.
- All that might be really complicated.
- You could have written variance-- well, anyway.
- I just want to show you that when you do the statistics
- class, it'll introduce you to a lot of things, but at the
- end, all you care about is the average
- distance from the mean.
- And I'll show you how easy that is to calculate.
- Everything I've done so far this problem you would not
- have had to do on a real exam.
- I'm just trying to give a little bit of intuition.
- All right, so the distance from 4 to 7 is 3.
- So the distance from data point one is 3.
- The distance from 13 to 7, what's that distance?
- Well, that's 7.
- We just care about the absolute distance.
- So then that distance is 7.
- The distance from 5 to 7 is 2.
- The distance from 6 to 7 is 1.
- The distance from 9 to 7 is 2.
- So we just want to take the average of
- all of these numbers.
- Let's see, 3 plus 7 is 10.
- 10 plus 2 is 12.
- 12 plus 1 is 13.
- Plus 2 is 15.
- So the sum of these is 15.
- And I have 5 data points.
- So divided by 5 is equal to 3.
- So the standard deviation is equal to-- well, that's
- strange because I got 3 and they have an answer, 3.3, so I
- feel like maybe I made a mistake.
- But, I mean, I'm going to go with that.
- I'm going to go with the 3.3.
- Maybe, I guess the actual mean is probably not 7.
- So if you used the non-actual mean you would have-- let's
- see what the actual mean is.
- And once again this is not something that you
- would have to do.
- Well, anyway.
- Actually, I just realized I've run out of time.
- So I'll continue this in the next video.
- See you soon.
- The next video is going to be one problem.
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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