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Question

A laboratory animal may eat any one of three foods each day. Laboratory records show that if the animal chooses one food on one trial, it will choose the same food on the next trial with a probability of $50 \%,$ and it will choose the other foods on the next trial with equal probabilities of 25$\% .$
a. What is the stochastic matrix for this situation?
b. If the animal chooses food $\# 1$ on an initial trial, what is the probability that it will choose food $\# 2$ on the second trial after the initial trial?

Amy Jiang · Accepted Answer

Okay, So have a P matrix of points wide 0.25 What you find to five here? Fine. Why put too fine. You point your fight. So let&#x27;s say this Executive One two on three. Okay, Now we&#x27;re asked. We were given our state victor, our initial vector, which is one no one we want to put out. We get quite five 0.25 important to flight. Okay, now let&#x27;s multiply. P times are in a director, Our little dish x one and it&#x27;s multiply P Temper X one back there. That gives us what? Why, What if I point you flying kind here and put fire rest of fortune five everywhere. Okay, It was an excellent actor. That was 0.5 point 25 0.25 And that just gives me play. 375 0.3125 0.3 once in flight through. So the probability of getting are true. The food three is 30.312

Abigail Martyr · Answer

in this problem. It&#x27;s talking about the food consumption of a certain a young animal. I suggest that you read the full problem inside the text, get the full details. What is important to note hair is that any people toe 1 20 on P is equal to 0.6. And we asked the find the probability that the animal consumes exactly eight units of food apart A and important be the probability that the animal consumes at least 17 units of food. We are going to use the by no the normal approximation to the binomial distribution. So first we will need to find the mean understand a deviation. So do mean is given by N Times P, which is 72 and standard deviation. It&#x27;s square root and times p times one minus is that gives approximately five point 37 The first. We wouldn&#x27;t know the probability that exactly ill units of foods are consumed, and that corresponds to the area beneath. I mean, if the bar of our history Graham from X equals 79.5 two x equals 80.5. The correspondent These cores we wanted records x minus the mean depended by the standard deviation. And that was breakfast lately. 1.4 Cyril the second z score. I need it. X value minus the mean read it by 5.37 on that day, 1.58 in front of probability. That we have X than Monta put Consumed is exactly 80 is equal to the probability that he&#x27;s less than 1.58 minus the probability that she is less than 1.4 from the table. Because the table we get point nine for 29 minus 0.91 Knight to and that gives us a probability of 0.2 37 Okay. Now, for the probability that the animal consumes at least 70 units of food so useful correspond to the area under the on the areas over our the bars of our history. Graham, beginning at 69.5, were extra 69.5. So we will find a corresponding the value here, physical to 16 or import five minus the mean good matter by the standard deviation from God. Approximately negative. The whole 0.47 You&#x27;ve heard of probability that at least 70 units were consumed equal to one, minus the probability that the less than negative 0.47 the one minus from the table we have 0.31 nine to gives a probability of point. Big A No A.

Kimberly Moon · Answer

Okay. So from our scenario, it states that there are 20 vegetarians, of which they state that nine of them eat both fish and eggs. Okay, then they tell us that three eat eggs, but they do not eat fish. No fish on lee eggs. Then they say that eight eat neither, So this group will not eat fish. Four eggs. The question actually asked for the probability that the probability of them eating neither. Based upon that, I&#x27;m talking about a proportion of my total. Well, my total is 20. And of those 28 would be our number that eat neither So eight out of 20 which equals 200.4 would be the probability that eat neither fish or eggs.

Sophie Knight · Answer

So Ron Problem 87 its has two parts Onda. We are told that the probability of a There&#x27;s no 0.5 and we&#x27;re also told that the conditional probability okay, because given because there&#x27;s no 0.25 on the last bit of information we get is that the conditional is that because given acres there&#x27;s no point to suppose a asked us to work out polled Rosie that because And this would need to do in two parts using both conditional formulas. So poverty be given a equals probability? Well, a on B over the probability of a Now we know two of the three things Onda we need. We&#x27;re trying to figure out poverty of A and because we&#x27;re gonna need that to account for pretty a bit. So the probability we just need to rearrange of being a be given a times probative eggs always on here is multiply both sides by poverty. A gives us that poverty of a that would be which is what we need. I mean, his property a big of a night. There&#x27;s no point I&#x27;m gonna times that by that no point five and that gives us no point Mom, I&#x27;m gonna write that. Okay, um equals no appointment. Now if we go to the other conditional formula So we&#x27;re just stopping the A&#x27;s and B&#x27;s around, we no problem to a given big equals formative A be over the public you beat. Now this is the equation that is gonna give us what we&#x27;re looking for. So rearranging when it&#x27;s times both sides by the probability of be on divide by the probability of a given B which gives a final equation the pope amenity of being equals poverty, a on B over property, a given bay. And we know a and B now, which is not put, Mom, when we&#x27;re divided that but a given be, which is no point to size. And that comes out as no point for so we know no problem. It&#x27;d be there&#x27;s no point Well, and thats the chance of it. It&#x27;s now maybe under part B is asking us what is the conditional probability that B does not occur given that a does not occur. So the formula to this would be Kober c of not a on Do not be which isn&#x27;t the same Azaz. Something about the hair probability of no A and not be is not property. Not am Be as important there. Because if you did if did this one you get an answer. Probably bigger than one, which obviously can&#x27;t be the case. And then we need wounded. Divide that further. Primitive. Not Okay. So work appropriated not a We need to do one monitor for Where&#x27;s the That is okay. And if you look back up A is no 0.5 probability of any The poets have not A is also no point fun. So we now know wanting for this equation. We&#x27;re also to work out the problems we have not a and not be going to need to know probability Uh, not fish. So provocative, not be is one minus probability of B. And we just went up being the previous question as no point for so that there&#x27;s no 0.6 now to get the probability of a no a no. B, not a and not B. We need to multiply. Problems have not a by the probability of not be, which gives us no 0.5 times no 0.6, but is no 0.3. So every now have everything we need for the equation. So the probability the B doesn&#x27;t occur given a doesn&#x27;t, uh, it&#x27;s polarity of not a and not B, which we now know is no 0.3. Andi, we&#x27;re dividing that by the poverty of not a which is not 0.5, which gives us no All right, Suze.

Robin Corrigan · Answer

for this problem. We&#x27;re told that 2% of two million high school students would take the S A T every year, requires special accommodations and then were asked to consider a random sample of 25 students who have recently taken the test. And we&#x27;re asking probabilities so weakened we can define X as the number of successes being the number of students who required special accommodation and because we&#x27;re sampling without replacement from the population. The samples are not strictly independent, but because 25 is far less than 5% of the total population, X can be pretty reliably estimated as a binomial random variable based on 25 size of 25 and the probability of success of 0.2 part A were asked, What is the probability that exactly one required special accommodation? What is the probability that we have exactly one success? So that&#x27;s the probability mass function for the binomial, and this comes out to approximately zero point 308 now for B were asked, what is the probability that at least one received special accommodation? So what is the probability that we have at least one success we can rewrite this as one minus the probability of getting exactly zero successes, and this comes out to approximately zero point 397 or C. We&#x27;re looking for the probability of at least two successes, so this could be rewritten as one minus probability. That X is less than or equal to one, and this equals one minus 0.911 which equals 0.89 for a party, you were asked, what is the probability that the number of successes is within two standard deviations of the mean number of successes or the expected number of successes? So to calculate this, we must calculate the mean number of successes where this distribution and the standard deviation So for a binomial random variable, the mean or expected value for the number of successes is he going to end terms? P, which comes out to you 1/2 and the standard deviation of a binomial random variable is given by the square root of N times. P Times Q, which comes out to 0.7, so within two standard deviations of the expected value. We&#x27;re within this range, so the mean value is 0.5 and so plus or minus two times 0.7, which is the standard deviation, and that gives us a range from minus 0.9 to 1.9. So that means we&#x27;re looking for the probability of a certain number of successes. So we for outcomes we can only have integer numbers of students, and that can&#x27;t be negative. So this would be replaced by zero Can&#x27;t be any lesson zero, and the number two would be more than two standard deviations greater than the expected number. So therefore, we must look for outcomes that are less than or equal to one. So we&#x27;re looking for the probability that the number of successes is between zero and one, which is the same as saying we&#x27;re looking for the probability of at most one success, which comes out to 0.911 and finally for party. It&#x27;s explained to us that students who receive special accommodation are allowed 4.5 hours for the S A T, and students who do not are allowed three hours to rate the S A T. So we&#x27;re asked, what do we expect? The average time allowed for the 25 students to be, so we can begin by writing a function that defines the average time spent by the 25 students. So for a student who is given special accommodation to get 4.5 hours, so we have 4.5 hours times number of students who get the special accommodation and then the students who don&#x27;t get three hours and the number of them will be 25 minus the number of successes. And to make it an average, we must divide by the number of students, divide by 25. So this function defines the average number of hours that the students are given for the S A. T out of a sample of 25 and this can be simplified as 75 plus 1.5 x over 25. So for the question, we&#x27;re asked, what is the number of hours allocated to the students on average, that we expect. So what is their expected number for the average number of hours allocated to the students? So what is asking for us? The expected value of the function h of X. So we can say that is equal to the expected value of 75 plus 1.5 x over 25. And because of the linearity of expectation, we can factory with the 1/25 and rewrite the rest of it like this. Now we have 1/25 so the expected value of a constant is just that constant. So we have 75 plus 1.5 times the expected value for X. And we&#x27;ve already calculated that previously, as 0.5. So this comes out to 3.3 So are we expect that the average number of hours allocated to the students to the 25 students writing S A T will be 3.3 hours.

Robin Corrigan · Answer

for this question, we&#x27;re told that a bowl campaigns three kinds of chocolates, milk, dark and semisweet, and we&#x27;re basically given these three statements in the question. So all but 50 or Derek all but 50 are milk, and all but 50 are semisweet and then were asked some questions about the sample space. So to try to figure out how the sample space looks, we can create a Venn diagram now. One thing we know is that there are only three kinds of chocolates in the sample space. So the derrick, the milk and the semisweet so weaken, partition it into three separate spaces for each of those three types of Candies. Because there is no overlap, so there&#x27;s no Candies that are both dark and milk. You can either be a dark chocolate milk, chocolate or semisweet, so let&#x27;s live with these milk, dark and semisweet. And because there&#x27;s no other kind of candy in this bowl, we know that they feel the entire sample space. So part A asks how many Candies are in the bowl. So if we want to figure this out, we can look at these three statements and we&#x27;re saying identical. We&#x27;re making identical statements about each type of chocolate were more saying, There&#x27;s all but 50 are that type of chocolate. Which leads us to believe that the number of milk is equal to the number of Derek, which is equal to the number of semisweet. So if we if we run with this assumption for a moment, we&#x27;re saying that all but 50 our milk so therefore there&#x27;s 50 in dark and semisweet. But if there is the same number of Derrick as SEMISWEET, then we&#x27;re saying that there must be 25 dark and 25 semisweet. So if we look at statement to than, say, statement one and say all but 50 are dark chocolate, that means there must be 50 between milk and semi sweet, since we know there are 25 semisweet. Therefore there must be 25 milk chocolate. And you can validate this by confirming that all three statements are still true. And so for part A. Therefore, the total number of chocolates is m plus the was s, and that equals 75. Well, for part B, how many of each kind are in the bowl so we can say milk is equal to you 25. Derek is equal to 25. Semisweet is equal to 25 for part C. If if a chocolate has selected at random out of the bowl, what is the probability that it is milk? So the probability that is milk is equal to the number of milk chocolates in the sample space divided by the total size of the sample space, which is 25 invited by 75 or 1/3 and for D were asked, What is the probability that it is milk or dark chocolate? So the probability of No or Derek is the number that are milk or Derek. What about the size of the sample space? So the number that are milk or dark is 25. 25 equals 50 over the total number, so that equals to over three. And then, for part, he were asked, What is the probability that it is milk and dark chocolate, so probability that it is milk and dark chocolate. So from the question, we know that that a chocolate can only be milk or dark or semisweet, but it cannot be both milk and dark. So finding a chocolate that is both milk and dark isn&#x27;t an outcome that does not exist in the sample space, so that event does not exist. So events that do not exist in the sample space have probabilities of zero of occurring, so that probability is zero.

Problem

On any given day, a student is either healthy or …

Problem 2 Medium Difficulty

Answer

a. $P = \left[ \begin{array} { c c c } { .5 } & { .25 } & { .25 } \\ { .25 } & { .5 } & { .25 } \\ { .25 } & { .25 } & { .5 } \end{array} \right]$
b. the probability that the animal will choose food $\# 2$ is .3125

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Caleb E.

Samuel H.

Michael J.

Joseph L.

Vectors Intro

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a. $P = \left[ \begin{array} { c c c } { .5 } & { .25 } & { .25 } \\ { .25 } & { .5 } & { .25 } \\ { .25 } & { .25 } & { .5 } \end{array} \right]$b. the probability that the animal will choose food $\# 2$ is .3125

Linear Algebra and Its Applications

Caleb E.

Samuel H.

Michael J.

Joseph L.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Caleb E.

Samuel H.

Michael J.

Joseph L.

a. $P = \left[ \begin{array} { c c c } { .5 } & { .25 } & { .25 } \\ { .25 } & { .5 } & { .25 } \\ { .25 } & { .25 } & { .5 } \end{array} \right]$
b. the probability that the animal will choose food $\# 2$ is .3125

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications