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Complete probability (ileurein Conditional probahilil} Using Laplace transforms solve Y() Find unit norma ector the surface Tal the point (2.2.1). 'crgcnce. Fi...

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Complete probability (ileurein Conditional probahilil} Using Laplace transforms solve Y() Find unit norma ector the surface Tal the point (2.2.1). 'crgcnce. Find the Taylor series COSZE with Ihe centet ; Qand determine the radius of conv

Complete probability (ileurein Conditional probahilil} Using Laplace transforms solve Y() Find unit norma ector the surface Tal the point (2.2.1). 'crgcnce. Find the Taylor series COSZE with Ihe centet ; Qand determine the radius of conv



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Find the area of the shaded region. Then find the probability that a point chosen at random is in the shaded region. Assume all inscribed polygons are regular. FIGURE CAN'T COPY.

My friends. Let's take a look at this problem. So in our situation here we have a circle just like this. And what we noticed is that we have a triangle on the interior of our circle. And what we recognize about their strangle is that it is an equal lateral triangle. And that is a really special thing, because what it does is it allows us to see a relationship amongst our degree is. And so what that tells us is that this is 120 degrees because each one of these sectors is also going to be 1 20 Together they would be 360 degrees. And this is the shaded area that we're discussing. Our goal is to figure out the probability of landing in this shaded area. And so what we're looking for is we're looking for shaded area. Are probability is gonna be our sheeted area over total area. And so our job is to figure out the area of this segment of our figure. But first, what we really need to do is to define what the area of the total is as well as what the area of this whole sector is by itself. And then our goal is to subtract the triangle toe, find what's left over down here. So let's see if we can get after it. Our first step is to find our total area. So our total area, my radius, is going to be eight units and that means that my diameter, it was 16 units. So my total area is gonna be pi radius squared, which is going to give me a 64 pi. So that's my total area. Now what? I need to find us to find my sector area. So sector area is actually going to be 1/3 of 64 pi. I could multiply 64 pi by 1 20/3 60 or multiply by 1/3 but it's the same as dividing. So I'm gonna say 64 pie divided by three. And that's my sector area here and 64 divided by three is going to be 21 a third. So, in order to, um, avoid rounding Oh, let's just yet. Let's do this with 21.33 pie. There we go. And then what we need to do is we also need to subtract the area of the triangle. And so this is where things get a tiny bit challenging, and it's always best to redraw. So men agree, draw our triangle up top here so we can kind of see and I addressed what's going on here. And so if this is a 120 degrees, that means that half of this is 60 degrees, which is awesome news, because this becomes a 30 60 90 triangle. And so, if this is eight, what we know is that the small side is going to be half of that. That's gonna before and then are medium sized side down here is gonna be for square root of three. So this is also forced Greater three down here. And so now I need to find the area of the triangle. So Triangle area is gonna be 1/2 the base, so four and four is eight. So we have eight squared of three, and our height is for so 1/2 of eight is four and four times for 16. So this is gonna be 16 square root of three. And so what I'm gonna do now is to erase the strangle, and we're gonna go back here. And so I have sector area, which is 21.33 and then I need to write my segments area just the smaller piece, which is gonna be 21.33 pi minus 16. Screw to three, which is the area of this of the triangle. So then I can take my calculator, and I could say 21 point, um, 33 actually, let's go ahead and set up our equation, and then we'll do that. So my shaded total right here, my shaded total is going to be 21.33 pi minus 16 spirit of three. And the reason we want to go ahead and put this all together is because rounding multiple times causes us to lose. Um, the accuracy and our numbers. So this is why we're gonna set it up like this. And then we're gonna divide by 64 pi, which is our total area. So then now we're ready to take our calculator and we're gonna say parentheses. 21.33 pie minus 16. Square root of three. Close those parentheses on top, and then we're gonna divide and open up our apprentices again on bottom, and we're going to say 64 pi close those parentheses and we should end up with 0.195 which would be 19.5%. So approximately 19.5% of your entire circle is the shaded region I shot.

All right, friends, let's take a look at this problem. So I went ahead and drew our picture here. What we've got going on our three sectors that air 72 degrees in we have the segments of those sectors that are shaded. Um, in our case here, all of these sectors are 72 degrees, which means that we have kind of a Pentagon shaped, um, happening inside of our figure. So what we also need to think about is that we're looking at the shaded total over the total area of the entire figure here. So my total area So my circle total area, There we go. Total area is going to be pi radius squared. In our case, 7.5 is the radius of our figure, which leaves us with 56.25 pipe. Now, will we Also need to do is to talk about this sector area, so sector area is going to be 56.25 pi and we're just going to divide that by five. And the reason why is because all of these sectors are the same amount and this is 72 degrees is 1/5 of our figure. So we have 56.25 and we're gonna divide that by five sweep 11.25 pi. Now, the reason I only I'm talking about one of these sectors is because my goal here is going to be to find one of these sector areas here, find one segment and then we'll multiply one segment by three essence. We have three of them here. So let's talk about how to find the area of this segment. We need to take the sector minus the triangle. So my sector here looks similar to this. And if this is 72 degrees, then we recognize that half of that is going to be 36 degrees. So this is 36 degrees here, and my radius is 7.5. And so what I'm going to do here with that is then used my trigonometry to find the in for the rest of our information here, I need to find both X and why here in order to find the area of my triangle. So let's go ahead and solve for X first. Hopefully, you're thinking back to your trigonometry days. And so from your 36 degree angle Here, this is your opposite. This is your iPod news, and this is your adjacent. So if we're going to find the X first Harmony was opposite hypothesis, which is going to be signed. So I'm going to say sign of 36 degrees is equal to X over 7.5, and then I'm gonna multiply 7.5 to both sides. So I got 7.5 times the sign of 36 that's going to give me 4.4. So I've got 4.4, and that is this length right here. So 4.4 and then I also need to find why. So this is going to be the adjacent in the high pot news. And so our Jason and high Pot news is going to be co sign. So we're gonna say co sign of 36 is equal to why, over 7.5 and then I'm gonna multiply 7.5 both sides again. But the summary news co sign so co sign gonna be 6.6 So 6.6, there we go. Actually around that appropriate, it'll be 6.7. It's around carefully here. So at six point seven we go, Oops, he's, uh, area. So this is going to be 6.7. So now what we're gonna do is be ready to find our area. So the triangle area triangle area is going to be 1/2 the base, which is 4.4 times to sort of 8.8, 4.4 times to me. 8.8 and then my height, which we just found was 6.7. So we got 0.5 times 8.8 times 6.7 There we go. And that's many of us. 26.7. So this is gonna be a 26.7. And then what we know is that our seg ment area So I'm gonna do this in blue. So seg ment area is going to be 11.25 pi minus 26.7, which is the area that triangle in what we know here. So 11.25 high minus 26.7 is going to have us 8.64 So 8.64 There we go. That is going to be our seg ment area. However, we're gonna multiply that times three because we have three of them. So 8.64 times three is gonna give us 25.93 So 25.992 Excuse me. So 25.92 And this is your shaded area. This is your shaded area. However, we're not quite done because we still need to be able to find the probability of landing in that shaded space. So we're going to say, go back to our kind of main idea here, and we're gonna say 25 0.92 divided by my total area, which is 56.25 pipe and let's go ahead and cargo that out. So I've got 25.92 divided by 56.25 pi, and that is gonna give me 0.1466 which we're gonna say is 14.66%. There you go. So 14.66% chance of landing in the shaded space. Excellent job

In this problem were given a formula to find the probability that you're going to hit a certain area on the dartboard. And in this one, we're trying to hit this area described right here. And if you look over on the dart board, you can see I colored it in blue right there. They told us everything that we need so we'll just go ahead and plug it in. Uh, data is going from nine pi over 20 toe. 11 pi over 20 are is going to three and 3/4 which is 15. Force 24 e to the negative x squared minus y squared. That's negative r squared and then d a which is R D r d theta So we have eat to the you. So if you is minus r squared, then d you would be negative to our d. R. So we need a negative two in there. We'll put a negative, too, and we'll put a negative one half out in front. So we get negative one half integral. Integrate this. You get e to the negative r squared from 4 to 15 force. Did they them? I'm pied over 2011 pi over 20. Okay, so notice that when you plug those numbers in, that will just be a constant. So you really just now need to go ahead and integrate the D theta, which will give you data. So you end up with this big constant here, minus one half E to the minus 16 minus E to the minus to 25/16 then the integral of Death Data's data. And that's supposed to be 11 pi over 20 minus nine pi over 20. Plug that into your calculator. You get 0.3 34 or 3.34 times 10 to the minus eight, which is not a very big probability.

Yeah, in this problem were given the formula for the probability of hitting a certain area on the dartboard. And in this one, we're interested in this blue area. Appear by the 20. So we'll set the problem up. It's one over. PI data is going from nine pi over 20 toe. 11 pi over 20 are is going from three and 3/4 which is 15 fours. 24 e minus X squared minus y squared is minus r squared and then d a is r D r D data. So we want to do eat to the you. So if you is minus r squared, then d u is minus two r d r So we need a negative two in there, which puts the negative one half on the front. So now we have negative 1/2 pi and a girl eat to the U. D you so it's in a gorillas eat to the U so e to the minus r squared from 13 force to four. Do you say that? Okay, so flood those numbers in and now you have negative 1/2 pi integral. Eat the minus 16 minus e to the minus 2. 25/16 d theta. So that's just a constant. So it can come out here with the 1/2 pi. And now all we have to do is integrate D theta, which will give us data from nine pi over 22 11 pi over 20. So when you plug those in, you get two pi over 20 and then that too pie in the top of that fraction cancels with the two pi at the beginning. So you end up with negative 1/20. Eat the minus 16 minus E to the minus 2 25/16 and put that in a calculator and you'll get 0.334 or 3.34 times 10 to the negative eight, which is not a very big probability.


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