Use Chebyshev’s inequality to show that a random graph G∈G_(n,p) almost surely does not contain a copy of K_4 if p≪ n^(-2/3).
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- Say you have a bipartite graph between X and Y with no edges, if N edges are added between X and Y uniformly at random what is the probability that the N edges forms a perfect matching.Find the maximum n such that the probability of getting no head in tossing a coin n times is greater than 0.1.Verify through listing all the possibilities of random graphs with three nodes that the probability that a graph with -nodes is connected formula is correct when n=3
- From a lot containing 2 % of defective products 400 items were drawn without returning them. Using Chebyshev's inequality, estimate the probability that among the drawn items the number of defective items will not exceed 5 %.Four apples and four oranges are distributed between two boxes in such a way that each of them has four fruits. At each step, we withdraw one fruit from each box and exchange them. Let Xn be the number of apples in the first box. Find the limiting probabilities.Color blindness appears in 1% of the people in a certain population. How large must a sample be if the probability of its containing a color-blind person is to be .95 or more? (Assume that the population is large enough to be considered infinite, so that sampling can be considered to be with replacement.)......
- What is the least value of k in Chebyshev’s Inequality for which the probability that X assumes a value greater than μ + kσ or less than μ − kσ is less than 0.05?Suppose Martin is a very talented used-car salesman. Whenever Martin talks to a new customer, there is a 90% chance that he convinces the customer to purchase one of his used cars. Brian, Martin's boss, is envious that Martin sells many more cars than he does. Because of his jealousy, Brian institutes a new rule that Martin is only allowed to talk to 35 customers per day. Thus, Martin continues to work each day until he speaks to 35 customers, at which point Brian sends him home. Let ? represent the number of used cars that Martin sells on a given day. What are the mean, ?, and variance, ?^2, of ?? Please round your answers to the nearest two decimal places. ?=_______? ?^2=_________?A sample of 51 planes will be taken from an infinite population. The population proportion equals 0.85. The probability that the sample proportion will be between 0.9115 and 0.946 is
- 4 A machine consists of 300 independently operating components, each of which can fail with probability 0.6. Using Chebyshev-Beynymè's inequality, estimate the probability that the frequency of failure will deviate (in absolute value) from the probability by less than 0.05.Suppose that arrivals to a coffee shop follow a Poisson process with average rate of one customer every 5 minutes. What are the probabilities of (a) no customers in 5 minutes; (b) exactly one customer in a minute; (c) exactly two customers in 2 minutes; and (d) at least two customers in 10 minutes?Draw the transition probability graph Pliskin and Tell (1981). Patients suffering from kidney failure can either get a transplant or undergo periodic dialysis. During any 1 year, 30% undergo cadaveric transplants, and 10% receive living-donor kidneys. In the year following a transplant, 30% of those who undergo the cadaveric transplants and 15% of living-donor recipients go back to dialysis. Death percentages among the two groups are 20% and 10%, respectively. Of those in the dialysis pool, 10% die, and of those who survive more than 1 year after a transplant, 5% die and 5% go back to dialysis.