Tom and Alice work independently in an attempt to solve a certain problem (i.e. whether one of them solves it does not affect the chances that the other will solve it). The probability that Tom solves the problem is 0.25 and the probability th solves the problem is 0.55. (a) What is the probability that the problem will be solved by at least one person? (b) If you find out later that the problem has been solved by at least one person, what is the conditional probability that Tom solved it?

Use the appropriate table to solve this problem: 30% of hedgehog owners choose fresh food to feed their pets (the rest chooses dried cat food). If 8 hedgehog owners are chosen at random, what is the probability that at most 2 of them feed their pets fresh food?             Fill in the following list:        n =                    p =                   Then work the problem:            (give the answer to four decimal places)

If Claude is happy one day, the probability that she's happy the next day is 1. If she's sad one day, the probability that she's sad the next day is 0.2. Over the long term, the probability that she's happy on any day is 1 (Hint: you can answer this using logic and maybe a drawing, no equations needed.) If Hau-Tieng is happy one day, the probability that he's happy the next day is 0.4. If he's sad one day, the probability that he's sad the next day is 1. Over the long term, the probability that he's happy on any day is 0 (Hint: you can answer this using logic and maybe a drawing, no equations needed.) If Suraya is happy one day, the probability that she's happy the next day is 0.7. If Suraya is sad one day, the probability that she's sad the next day is 0.6. Over the long term, the probability that she's happy on any day is 57.14 (Hint: time to write down a transition matrix and use Markov Chain tools...)

Chapman-Kolmogorov EquationsSuppose that a communications network transmits binary digits, 0 or 1, where each digit is transmitted 10times in succession. During each transmission, the probability is 0.995 that the digit entered will betransmitted accurately. In other words, the probability is 0.005 that the digit being transmitted will berecorded with the opposite value at the end of the transmission. For each transmission after the first one, thedigit entered for transmission is the one that was recorded at the end of the preceding transmission. If X0denotes the binary digit entering the system, X1 the binary digit recorded after the first transmission, X2 thebinary digit recorded after the second transmission, . . . , then { Xn} is a Markov chain.  A. Construct the (one-step) transition matrix. B. Find the 10-step transition matrix P(10). Use this result to identify the probability that a digit enteringthe network will be recorded accurately after the last transmission. C. Suppose…