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(1 point) Trees planted by a landscaping firm have 90% one-year surviva rate , If they plant 16 trees in a park what is the following probabilities:1 All the trees ...

Question

(1 point) Trees planted by a landscaping firm have 90% one-year surviva rate , If they plant 16 trees in a park what is the following probabilities:1 All the trees sunvive one year: Answer:2. At least 14 trees survive one year: Answer:

(1 point) Trees planted by a landscaping firm have 90% one-year surviva rate , If they plant 16 trees in a park what is the following probabilities: 1 All the trees sunvive one year: Answer: 2. At least 14 trees survive one year: Answer:



Answers

Of all the trees planted by a landscaping firm, $90 \%$ survive. What is the probability that 8 or more of the 10 trees they just planted will survive? (Find the answer by using a table.)

So this is a binomial distribution problem. We are gonna have to do more than one binomial distribution and then be able to add up all the probabilities at the end to be able to solve this out. But we'll work through that. So we know what we need to be able to fill out to solve out a binomial distribution is we need in our number of trials. We are a number of successes. We need P R. Probability of success on any given trial and cue a probability of failure in any given trial. So, looking what we've been given, we're told that 200 shrubs air being planted so that would represent our end que en is going to be 200. That's a total number of trials. We're told that the probability of failure to grow is 0.2 Now I recognize that this is the probability of failure. But understand is the probability of failure to grow. So it's just saying in the world of our question here where we're planting shrubs and we're deciding whether they grow or failed to grow, there's a 0.2 chance that they will fail to grow we don't know if that is R, P or R. Q. Until we know what we're actually trying were being asked about. Okay, it is asking, was the probability that three that at most three will need to be replaced? Well, if we are replacing them, that would imply that they have failed to grow right. You wouldn't replace a show that is growing the way it's supposed to. So if we're asking about the probability of ones that need to be replaced, as weird as it sounds, our success is when a shrub fails to grow. They'll say that again, since we're being asked about the probability of which shrubs need to be replaced. If we care about the ones that are being replaced, that's our successes. When I show up has to be replaced. We could. We consider that a success, and the ones we replace are the ones that fail to grow. So the ones that failed to grow counters our successes for this question, which means P our probability of success is actually 0.2 que. Then our probability of failure would be 0.98 because remember our probability of success. Suppose our probability of failure. I should add upto one or another way of saying that is one minus. P should give us Q one minus. Q. Should give us p So if we take one minus 10.2 we get 0.98 which would be our Q. In that case, okay, are is going to be different, depending on which one were doing okay. It says at most three can be replaced, so three is certainly acceptable. But that's the most that we can replace. So nothing mawr. Nothing higher than three is acceptable. It is acceptable if there is less than three. Meaning to replace shoves is okay. One replaced shrub is okay. And replacing zero shrubs is also okay. So we have four different probability distributions that we will have to do and then we can add up all four of those at the end. So we need to find the probability of replacing zero shrubs. We need to find the probability of replacing one trip. We need to find the probability of replacing two shrubs, and we need to find the probability of replacing three strips. All right, so we'll start with zero total number of trials are N 200 total number of successes that we're looking for. Zero. We want zero successes that is being multiplied by 0.2 That's our probability of success, right? But that is to the zero power, meaning it's kind of useless and pointless because we don't actually want anything to fail. We don't actually want to have to replace any shrubs just exceed. In this case, probability of failure is 0.98 and that sorry bad princes. And that needs to be taken to the 200 minus zero power, meaning the 2/100 power. Right? So like we've said before, you can take all that and plug it into your calculator as is, or you can. If you have a calculator with a binomial pdf function, then you can go ahead and plug it into that. Either way, if you plug this in correctly, you should get that the probability of zero shrubs being replaced, meaning the probability of zero successes would be 0.0 17 and we'll around it to a 6175879 and blah, blah blah, so we'll just cut it off it. Four decimals came so we'll say 0.176 for the probability of one trip. Still 200 total trials. The only thing that's changing is now we want one success. I'm still multiplying 0.2 by 0.2 for our probability of success. But now it's actually the first power, so we really would be using it. We've still got 0.98 for a probability of failure, but now will be 200 minus one, meaning we're taking it to the 199th power, which doesn't seem like a big difference from 200 power. But that will change our problem or change our probability. I should say so again. You want to put that on your calculator one way or another, Feel free to pause the video so that you can take the time to plug it in. That you need. Once you plug it in, you should get 0.71 and again, we'll have to round and will be rounding up to an eight. Yeah, so then we go to two ships again. 200 trials, 200 shrubs still, but we want to replace two of them still have a probability of 0.2 as far as our success is concerned. But now would be the second power still have a probability of failure of 0.98 But now we're taking it to the 200 minus to so plugged that into your calculator or plug, you know, playing with your head clear, one way or another. If you plug this incorrectly, you should get 0.14 58 Okay, Last but not least, we need to solve out the probability for three ship shrubs. So 200 shrubs, 200 trials and then three possible successes. Then we need to multiply that by 0.2 again. But this time to the third power. I think you guys kind of have this down by now. Then times 0.98 to the 200 minus three. Change color My street This time again, feel forget positivity. If you need to do go toss all that into your calculator and such, or if you're ahead of me. Sorry. You should be getting zero point one 963 Now all we need to do is add up all four of those probabilities and we should be able to get our final answer. So we need to take 0.176 plus 0.718 plus 0.1458 plus 0.1963 free. Adul those up. Together, we should get a final probability of 0.4315

This question we have been given that X is, ah, variable where the number off trees planted in Sandy Oi. So this for the sandy soy that's so live one year and they've given the probability off exonerate that s p X, and that is given as to your 10.7. And then they have also given us not only able why, which is for the number of trees planted in a day that survive one year. So this is when they have planted in a day and the probability for this is given asked 0.6 and the total number off values are given to us as, ah, 50 each. So I would therefore take n oneness for X and end to us for by So what off them are equal to 50. Now I'm going to use a concept off by a normal distribution or here so the binomial distribution can be written. But the variables as first just for X with the number of fairly, was being 50 and the provided he is given as 500.7 and the next is supported by the number of values is 50 and the tablet he's given us 56 So the step you're now is to find the expectation off X minus y. So you're in the legality property. We can say that this will be given by your ex minus your Why which is equal to we have the oh expectations for except by is what we're supposed to first find here so far by normal distribution expectation off X, the formula will be given by N one indoor pX so this would be equal to 50 in Toby excess 0.7. So if you multiply, we're getting new eggs, which is the mean for X is equal. Do tow five. Next is to find the expectation off by so again for the binomial distribution expectation off, I will equal do n do in tow Be vile. So this is 15 in tow. Be rise 0.6. So we get the expectation of eye, which is nearby, is equal to 30. And now if I substitute this values you're in expectation off this formula X minus y. Then you can say you off X minus Y is equal. Do mew X minus mu. Why? Which is equal to 25 minus 30 which is equal to five moving further. We're supposed to find out of aliens off X minus y also. So to find the radiance of X minus y you want to use the formal affiliations and the linearity properties that will be given by you off X and minus one square will change to plus one on me every off Why, But we need to first find the millions of eggs and billions off Why so far by normal distribution, billions off ex can be found using the formal off end one B X and into one minus B x. So this will be good. 15 in Toby excess 0.7 and into one minus 10.7 is 0.3. So if I might apply the value, I'm getting your four millions off exes in 40.5. Next is to calculate radiance off. Why? So this will be given by end to Indo be vice and into one minus beaver. So substituting the values this is 50 into this is wind. Six into one minus 10.6 will be wind ford. And if you might apply all the three domes, then the readings off. Why I'm getting here is well, so now continuing with finding volumes off X minus y. This is equal. Millions off express millions off. Why so millions of excess? 10.5? Those readings off by a stray. So we add and begin the value. Here is 22.5 Now. Using this, we want to find a standard deviation for X minus way, just given by squared off regions. So we just rattled off 22.5, and this value am getting as 4.74 Now we've got to get to the main parameters for X minus y. Now you go to simplify the main question that the last if told us to find compatibility off minus phi, there's an equal X minus y there's an equal five. So to simply find this part, we've captured all the parameters of X minus y already. So I will use the formal off cell, as in place off xbm X minus y someone to use that we have X minus wife minuses. Mule off X minus Y and divided by standard deviation off X minus wife. So there are two values of X minus way at the value off X minus Y is equal minus five. We didn't get filled the value off they So this is minus five minus five, Divided by sigma. It's minus y is 4.74 So doesn't really that getting hurt is minus 2.11 Next, at X minus, Y is equal plus five billion Catholics there, which is equal do five minus 514.74 So this is equal to zero. So, in terms of that, this probably therefore this probability off minus 2.11 That's an equal to sell. There's an equal to zero. And this scandal Dennis fire off zero minus phi off minus 2.11 So from the standard normal distribution table, we can take these values. So far, zero we have the value from the devil s 00.5 because this is a DEA off standard normal called. So left half right half off. The key area is 0.5 minus from the table minus 2.11 will give the value the you know, 0.174 So if I cancel it, then the value of getting her 0.48 46 So this is the final answer for the question that doesn't ask

All right, we are going to first think of this problem as a realistic situation. That's how I like to think about stuff. So let's say we've got this logging company that we're working for and they're wanting to know how many trees there are in this forest. What they give you is that there's 80 trees per acre on average, or that's what they're expecting. On average is about 80 trees per acre. So let's first, right. What we know. Yeah. So we're going to be using the poison distribution probability function. You land a, uh, bye. And it's X over ex factorial we are. We know that Lambda equals 80 trees. Her acre. Okay. And we don't know the probability of X. So you share. Expect this is our probability. All right, a right. So on a what's right, what we know. So what we know is we've got a quarter acre is what we're focusing on, which is equal to a quarter, and we want to know x from one to 16. We can't do zero because the probably probability distribution uh, boys on distribution probability function does not do zeros. It's always from one to something So what's figure out how maney trees we need. So we've got our quarter acre times one acre, four trees equals 20 trees. Not will be Orlando Lambda. Okay, so let's put that in the points on distribution Probability function 20 for Lambda, one through 16 for X, which will be a sum equation. Land equals 20 and we have X, which equals one to 16 of E to lambda Times. Lambda the X over X factorial. Now, from here, since we're doing statistics, I go to excel, right? So in excel Ah, you should do most your math and itself for statistic problems, we have land at 20 for each X, we have from 16 to 1 as shown, uh, this is our equation. This is each answer for each X, and then we sum it all together to get our answer. So this Ben equal 0.2 to 1. All right, let's go on to be be we have what we know, which is there is 85,000 acres in the forest. What, and what we want to know is how many trees were expecting toe have in that forest. So let's do the math. So we have 85 1000 acres. Forest times. We have 80 trees to one acre. That gives us about six million 800,000 trees for the forest. All right, that's B. So let's work on C. Right. So, for C, we write what we know. Uh, which is? Yeah, radius of 0.1 miles. And we have one square mile equals 640 acres. And what we want to know is the PM f of X. So let's try and figure out what our lambda is. So, first off, we want to know what are the square miles in that circle. So that's pretty easy for square miles would be equal to. Yeah, pliers. Where'd equals bye to the 0.1 squared. So we get one. Hi. Next we try and figure that out for Lambda Lambda equals 0.1 pie times 640 acres per square, mile, times or 80 trees. So then we will get 500 12 hi. Which is about roughly 1006 108 point 495 But I like to work with exact, so we're going to figure out the P M f of X. She calls p of 512 pi, which is our Lambda X equals E? Yeah. 512 pi. Okay, Class 12 bye over X or to the X over ex factorial. And that is your answer because they don't really give you what X is. So that's C. This is B. And this is a uh huh. Thank you.

For this problem, I would like to compare it to a logging company. This logging company wants to know how maney trees there are in a forest as well as how many trees are they going to get from this forest? They first tell you that there's about 80 trees on average per acre, and they would like you to help them figure out what they can get from this forest. So let's first start by what we know. So what we know is we're going to be using the poison distribution probability formula. Lambda equals 80 trees per acre as what we expect and her ex, which we don't know, will be our probability or what we want. So for part A, what we know is that there's going to be a core acre, so that's gonna be 0.25 acre, and what we want is the probability of X from 1 to 16. So first starting this, we want to know how many trees we need, which would be our lambda. Our land death equals. What's your point 25 acre times 80 trees per acre and then the acres cancel out and we get 20 trees for that quarter acre. Next we're going Thio, use the poison distribution probability formula. So Lambda is 20. X is from one to 16. And for this we want to use the someone the summation of Lambda for 20 and X from one to 16. So this would be e minus e to the negative 20th times 20 to X all over Ex factorial, which is very similar to It's an egg of 20 times 20 the one over one factorial plus e to the negative 20 20 to the second all over two factorial. And this goes on until we get to e to the negative 20 to 20 to 16 all over 16. Factorial For this problem, I would highly suggest using Excel with Excel. Most statistic problems are used, uh, in statistics, you use Excel quite a bit. So it highlights to just using Excel on game used to using it with itself. We find that the solution is zero point 2 to 1 074 and that's our solution for party part B. What we know is there's 85,000 acres to the forest. What we want to know is the expected trees in the forest. This is a very simple problem. So we have 85 1000 acres to the forest. Times are 80 trees for one acre, the acres cancel out and we get a solution of six million 800,000 trees to that forest. So that's part B. I was very simple. Part C. What we know is there is going to be a circle in the middle of this forest, and that circle has a radius of 0.1 mile and for one square mile is equal to 640 acres. What we want is the P M F of X. So starting out, we want to know how many square miles there are in that circle. So to figure that out, we have pie R squared which is equal to pie to the point one squared, which is equal to hi. The point 01 This is our miles squared. Now to figure out our Lambda are Lambda is hi to the 0.1 mile squared times 640 acres for one mile squared times 80 trees, 21 acre. So the acres cancel out the mouse squares cancel out and we get 512 hi trees, which is about roughly 1608 0.5 trees. But I prefer using MAWR exact answers. So for PMF is our probability distribution formula or poison distribution probability formula which, well, BP to the 500 12 pie over X, not over x two with X, and we get e the negative 512 pie Times 500 12 pie two X over ex factorial This is as far as we can get with part C. So this is the answer for the PMF.


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