Suppose a jar has four chips numbered 1.2.3.4. Suppose a person draws a chip and

Question

Suppose a jar has four chips numbered 1.2.3.4. Suppose a person draws a chip and s another, returns it, and so on until he gets Problem 1. the first draw is always chip 3. He returns it, draw chip 3 the second time. 1. What is the probability that the person draws exactly 4 times? [5 pts 2. Model a Markov chain process for this random experiment. [10 pts] 3. What is the ex 4. On average, how many times will the person get chip 2? [10 pts pected total number of draws for this experiment? 10 pts Problem 2. Consider one specific type of John Deere mowers that Lowes sells. There is a war- ranty on all mowers that requires free replacement of any mower that fails before it is 3 years old. Suppose that (1) 2% of all new mowers fail during the first year; (2) 5% of one-year-old mowers fail during the second year of operation; and (3) 10% of all 2-year-old mowers fail during the third year (1) Model the sells of the mowers using a stochastic process by defining the state variable and transition probability matrix. [5 pts (2) Which states are transient, and which are absorbing states? [5 pts] (3) What is the fraction of all mowers that Lowes will have to replace? [10 pts (4) Suppose that it costs Lowes $300 to replace a mower and Lowes sells 7,000 mowers per year. If Lowes reduced the warranty period to two years, how much would be saved? [10 pts)

Explanation / Answer

1. Now the jar has 4 chips named 1,2,3,4. A person draws a chip, the first draw is always chip 3. He returns it and keeps drawing in a similar manner until the draw is again 3. Since the prob of drawing a chip in an unbiased way is 1/4, the probability that he draws exactly 4 times = probability of drawing 3 on the 4th draw.

Since this drawing with replacement prob of drawing 3 is always 1/4 on any particular draw, so probability of drawing 3 on the 4th draw = 1/4

3. Since each draw has probability of 1/4 as this is drawing with replacement the expected number of draws = 4

and most of the draws required in different expts will be clustered around this.

4. On average the number of times, a person will get chip 2 will be 1 time as the expected number of draws are 4 and probabilty of each draw = 1/4 so number of times, a person will get chip 2 =4*(1/4) = 1

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