Example: Picking a non-blue marble Example of figuring out the probability of picking a non-blue marble from a bag.
Example: Picking a non-blue marble
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- Let's do a couple of excercises from our probability one module. We have a bag with nine red marbles,
- two blue marbles, and three green marbles in it. What is the probability of randomly selecting a
- non-blue marble from the bag? So lets draw this bag here. So that's my bag. We're going to assume that
- it's a transparent bag. That looks like a vase. We have nine red marbles. So we draw nine red marbles.
- One, two, three, four, five, six, seven, eight, nine red marbles. That looks some kind of orangesh but
- does the job. Two blue marbles. So we have one blue marble. Two blue marbles. And then we have three
- green marbles. Three green marbles. Let me draw those three...One, two, three. What is the probability
- of randomly selecting a non-blue marble from the bag? We mix them all up,
- and we have an equal probability of selecting any one of these. And the way you just think
- about it is: What fraction of all of the possible events meet our constraints. So lets just think about
- all of the possible events first. How many different possible marbles can we take out? Thats just the
- total number of marbles their are. There are one, two, three, four, five, six, seven, eight, nine , ten,
- eleven, twelve, thirteen, fourteen possible marbles. So this is the number of possibilities. And then
- we just have to think what fraction of those possibilities meet our constraints?
- The other way you could have gotten fourteen was by taking 9 + 2 + 3. So what number
- of the possiblities meet our constraints. And remember our constraint is selecting a non-blue
- marble from the bag. Another way of thinking about it is a red or a green marble. So how many non-blue
- marbles are there? Well there's a couple of ways of think about it. You could say there is 14
- total marbles: two are blue, so there are going to be 14 minus 2 which is 12 non-blue marbles.
- Or, you could just count them: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. So there are twelve non-blue marbles.
- So that's the number of non-blue, so these are the possibilities that meet our constraints
- over all of the possibilities. And then, if we want to- this isn't in lowest simplified... this isn't in simplified
- form right here, since both 12 and 14 are divisible by 2- lets divide both the numerator
- and the denominator by 2 and you get 6 over 7. So we have a six-seventh chance of
- selecting a non-blue marble from the bag. Let's do another one.
- If a number is randomly chosen from the following list, what is the probability
- that the number is a multiple of five? So once again, we want to find the fraction of
- the total possibilities that meet our constraint. And our constraint is being a multiple of five. So
- how many total possibilities are there? Let's think about that. Total possibilities... How many
- do we have? That's just the total number of numbers we have to pick from. So that's just 1, 2, 3, 4, 5, 6,
- 7, 8, 9, 10, 11, 12. So there are 12 possibilites. We have an equal chance of picking anyone of these 12.
- Now, which of these twelve are a multiple of five?
- So let's we do this in a different color, so let me pick out the multiples of five.
- 32 is not a multiple of five, 49 is not a multiple of five, 55 is a multiple of five.
- Really we're just looking for the numbers that in the ones place either have a five or a zero.
- 55 is a multiple of five, 30 is a multiple of 5, that's 6 times 5, 55 is 11 times 5, not 56, not 28,
- This is clearly 5 times 10, this is 8 times 5, this is the same number again, also 8 times 5,
- so all of these are multiples of 5, 45, that's a 9 times 5, 3 not a multiple of 5, 25 clearly 5 times 5.
- So I circled all the multiples of 5. So, of all the possibilities, the ones that meet
- our constraint of being a multiple of five, there are : 1, 2, 3, 4, 5, 6, 7 possibilities.
- So 7 meet our constraint.
- So, in this example, the probability of selecting a number that is a multiple of 5, is seven twelves.
- Let's do another one.
- The circumference of a circle is 36 pi. Let's draw this. So the circle looks like...
- I can draw a neater circle than that. So let's say the circle loks something like that.
- And its circumference, we have to be carefull here, they're giving us interesting... so circumference ...
- the circumference is 36 pi, then they tell us that contained in that circle is a smaller circle
- with area 16 pi. So inside the bigger circle we have a smaller circle that has...
- this guy right over here that has an area of 16 pi.
- A point is selected at random from inside the larger circle, so we're gonna randomly select some point in
- this larger circle. What is the probability that the point also lies in this smaller circle?
- So here's a little bit interesting, 'cause you actually have an infinite number of points in both of
- these circles, because it's ... it's not kind of separate balls or marbles, like we saw in the first example,
- or separate numbers. There's actually an infinite number of points you could pick here, and so
- when we talk about the probability that the point also lies in this smaller circle, we're really thinking
- about the percentage of the points in the larger circle, that are also in the smaller circle.
- Or another way to think about it is the probability that the point is also ... the probability
- that if we'll pick a point from this larger circle, the probability that's also in the smaller circle
- is really just going to be the percentage of the larger circle that is the smaller circle. I know that
- might sound confusing, but we really just have to figure out the areas for both of them, and it's just
- really going to be the ratio. So let's think about that: So there's a temptation to just use this 36 pi up here,
- but we have to remember: this was the circumference and we need to figure out the area of both of these
- circles. And so for area we need to know the radius 'cause area's pi*R squared, so we can figure out the
- radius from the circumference by saying, well circumference is equal to
- two times pi times the radius of the circle, or if you say 36 pi, which we
- were told this is the circumference, is equal to two times pi times
- the radius. We can divide both sides by 2pi , and on the left hand side,
- 36 divided by 2 is 18, the pi's cancel out... we get our radius is being
- equal to 18, for this larger circle. This larger circle has a radius of 18.
- So if we want to know it's area, it's area it's going to be pi*R sqared.
- Which is equal to pi times 18 square. Let's figure out what 18 square is.
- 18 times 18: 8 times 8 is 64, 8 times 1 is 8, plus 6 is 14, and then we have
- that zero there 'cause we're not in the ten's place, 1 times 8 is 8,
- 1 times 1 is 1, and really this is a 10 times a 10, that's why gives us 100.
- Anyway... 4 plus 0 is a 4, 4 plus 8 is a 12, and then 1 plus 1 plus 1 is 3,
- so thet's 324. So the area here is equal to pi times 324, or we could say 324 pi.
- So the area of the entire larger circle, the part that I shaded in yellow,
- including what's kind of under this orange circle, if you want to
- view it that way, this area right over here is equal to 324pi.
- So the probability that a point that we select from
- this larger circle is also in the smaller circle is really
- just a percentage of the larger circle that is the smaller circle
- so our probability... I'll just write it like this, the probability of ... that the point is
- also lies in the smaller circle .. so all of that stuff I'll put it ... the probability
- of that is going to be equal to the percentage of this larger circle that it is the smaller one
- so that's going to be... well... the fraction of the larger circle's area that is the
- smaller circle's area. So it's going to be 16pi over 324pi.
- And the pi's cancel out and both of them are divisible by 4 , if we divide the
- numerator by 4 we get 4, if we divide the denominator by 4, what do we get?
- 4 goes in 320, 80 times and goes into 4 once, so we get an 81.
- So the probability ... I didn't even draw this to scale, this area is actually much smaller
- when you do it to scale... The probability that you... if you are to randomly select a point
- from a larger circle that also lies in the smaller one is the ratio of the areas
- the ratio of the smaller circle to the larger one
- and that is four eighty ones, I guess is the best way to say it.
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