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let's do a couple of exercises from our probability one module so 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 let's let's draw this bag here so that's my bag and we're going to assume that it's a transparent bag that looks like a like a vase but let's we have nine red marbles so let me draw nine red marbles one two three four five six seven eight nine red marble so that looks like kind of orangish 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 so one two three what is the probability of randomly selecting a non-blue marble from the bag so maybe 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 is what fraction of all of the possible events meet our constraints so let's just think about all of the possible events first how many different possible marbles can we take out well that's just the total number of marbles there are so there are 1 2 3 4 5 6 7 8 9 10 11 12 13 14 possible marbles so this is the number of possibilities number of possibilities possibilities and then we just have to think what fraction of those possibilities meet our constraints and the other way you could've gotten 14 is just taking 9 plus 2 plus 3 so what what number of those possibilities meet our constraints and remember our constraint is selecting a non-blue a non-blue marble from the bag another way to think about it is a red or a green marble because the only non blue ones are only other two colors we have our red and green so how many non-blue marbles are there well there's a couple of ways to think about it you could say there's 14 total marbles to our blue so they're going to be 14 minus 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 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 this isn't in simplified form right here since both 12 and 14 are divisible by two so let's 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 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 total possibilities possibilities total possibilities how many do we have well that's just a number of numbers we have to pick from so 1 2 3 4 5 6 7 8 9 10 11 12 so there are twelve possibilities we have an equal chance of picking any one of these twelve now which of these twelve are numb our multiple of five so let's let me do this in a different color so let me pick out the multiples of five 32 is not a multiple 549 is not a multiple 5 55 is a multiple of five really we're just looking for the numbers that are in the ones place either have a 5 or a 0 55 is a multiple of 5 30 is a multiple of 5 that's 6 times 5 that's 50 5 is 11 times 5 not 56 not 28 this is clearly 5 times 10 this is 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 is not a multiple of 5 25 clearly 5 times 5 so I've circled all of the multiples of 5 so of all the possibilities the ones that meet our constraint of being a multiple of five there are one two three four five six seven possibilities so seven meet our constraint meet our meet our constraints so in this example the probability of selecting a number that is a multiple of five is seven twelve let's do another one let's do another one the circumference of a circle is 36 PI let's draw this let's draw this circle the circumference of a circle is 36 PI so I let's say the circle looks I can draw a neater circle than that so let's say the circle looks something like that and the circumference we have to be careful 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 has an area of 16 pi a point is selected at random from inside the larger circle so we're going to 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 because you actually have an infinite number of points in both of these circles because it's they're not it's not it's not kind of separate balls or marbles like we saw in the first example or separate numbers there's actually a infinite number of points you can pick here and so when we talk about the probability that the point also lies in the 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 were to 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 really just 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 because area is PI R squared so we can figure out the radius from the circumference by saying well circumference is equal to 2 times pi times the radius of the circle or if you say 36 pi which we were told as a circumference is equal to 2 times pi times the radius we can divide both sides by 2 pi divide both sides by 2 pi and on the left hand side 36 divided by 2 is 18 the PI's cancel out we get our radius as being equal to 18 for this larger circle this larger circle has a radius of 18 so if we want to know its area its area is going to be PI R squared which is equal to pi times 18 squared now let's figure out what 18 squared is 18 times 18 8 times 8 is 64 8 times 1 is 8 plus 6 is 14 and then we have we put that 0 there because we're now in the tens place 1 times 8 is 8 1 times 1 is 1 and really this is a 10 x a 10 that's why I guess it gives us 100 then anyway 4 plus 0 is a 4 4 plus 8 is a 12 and then 1 plus 1 plus 1 is a 3 so it'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've shaded in yellow including what's kind of under this ordinate that way this area right over here is equal to 324 pi so the probability that a point that is 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 in the probability of that is going to be equal to the percentage of this larger circle that is a smaller one and that's going to be or we could say the fraction of the larger circles area that is the smaller circles area so it's going to be 16 pi over over 324 pi over three hundred and twenty-four pi and the PI's cancel out and let's see it looks like both of them are divisible by four if we divide the numerator by 4 we get 4 if we divide the denominator by 4 what do we get 4 goes into 320 80 times it goes into 4 once so we get an 81 so the proud of probability so I didn't even draw this to scale this this area is actually much smaller when you do it to scale the probability that you if you were to randomly select a point from the larger circle that it also lies in the smaller one is the ratio of the area is the ratio of the smaller circle to the larger one and that is 480 once I guess it's the best way to say it