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- Let A and B be independent events on the probability space (Ω, ?, P). Prove that the complement of A (or Ac) and the complement of B (which is Bc) are also independent.Maui played a game of darts where a player gets to throw six darts to burst balloons. Basedon the performance of the previous customers, there is a 40% chance that a dart will hit aballoon. Let X be the number of balloons successfully hit by Maui. Define a function that willgenerate the probability of X. Show that it is indeed a probability function using the following axioms:P is a probability distribution on a sample space Ω, and let A, B ⊆ Ω be events with P(A) = 0.6 and P(B) = 0.7. Show that 0.1 ≤ P(B\A) ≤ 0.4.
- Pacific salmon populations have discrete breeding cycles in which they return from the ocean to streams to reproduce and then die. This occurs every one to five years, depending on the species. (a) Suppose that each fish must first survive predation by bears while swimming upstream, and predation occurs with probability d. After swimming upstream, each fish produces b offspring before dying. The stream is then stocked with m additional newly hatched fish before all fish then swim out to sea. What is the discrete-time recursion for the population dynamics, assuming that there is no mortality while at sea? You should count the population immediately before the upstream journey. nt + 1 = (b) Suppose that, instead of preying on fish while they swim upstream, bears do so only while the fish are swimming downstream. What is the discrete-time recursion for the population dynamics? (Again assume there is no mortality while at sea.) nt+1 (c) Which of the recursions obtained in parts (a) and (b)…Of all airline flight requests received by a certain discount ticket broker, 70% are for domestictravel (D) and 30% are for international flights (I). Let x be the number of requests among the next three requests received that are for domestic flights. Assuming independence of successive requests, determine the probability distribution of x.Let E and F be events in a sample space S with probabilities p (E)= and p (F) = such that the conditional probability of F given E is p(FE) = Determine p(EF). 0 / / 75 O None of these.
- Assume that the airline still sells 340 tickets for the 280-seat flight, but that all tickets are sold in groups of 4 and that each group of four only shows up with probability 0.854, since the entire group will only travel if all four in the group can travel. Compute the exact probability that there will be enough seats for everyone who shows up, noting that each party only shows up in full or not at all. Express your probability to three decimal places.Assume that Tom attends class randomly with probability 0.55 and that each decision is independent of previous attendance, so that the process can be viewed as a Bernoulli process. What is the probability that he attends at least 7 of 10 classes given that he attends at least 2 but not all 10 classes?A student gets selected by 53% to go on a science trip. Let X = the number of students who get selected in the next 6 students we have chosen at random. what is the probability that X is at most 1 O a. 0.989221 O b. 0.022164 O c. 0.083711 O d. 0.977836
- Pacific salmon populations have discrete breeding cycles in which they return from the ocean to streams to reproduce and then die. This occurs every one to five years, depending on the species. (a) Suppose that each fish must first survive predation by bears while swimming upstream, and predation occurs with probability d. After swimming upstream, each fish produces b offspring before dying. The stream is then stocked with m additional newly hatched fish before all fish then swim out to sea. What is the discrete-time recursion for the population dynamics, assuming that there is no mortality while at sea? You should count the population immediately before the upstream journey. nt + 1 = (b) Suppose that, instead of preying on fish while they swim upstream, bears do so only while the fish are swimming downstream. What is the discrete-time recursion for the population dynamics? (Again assume there is no mortality while at sea.) nt+1 = (c) Which of the recursions obtained in parts (a) and…The accompanying table shows the numbers of male and female students in a particular country who received bachelor's degrees in business in a recent year. Complete parts (a) and (b) below. Business graduates Business None degree degree male 181823 612083 female 166501 867055 • Find the probability that a randomly selected student is male, given that the student received a business degree. The probability that a randomly selected student is male, given that the student received a business degree, is I (Round to three decimal places as needed.) • Find the probability that a randomly selected student received a business degree, given that the student is female. The probability that a randomly selected student received a business degree, given that the student is female, is (Round to three decimal places as needed.)Consider a probability space (2, F, P) and a random variable X defined on it. Recall the definition of the distribution ux. Show that the distribution defines a probability on B.