Nicholas is competing against two friends in a series of races. Each day they ha

Question

Nicholas is competing against two friends in a series of races. Each day they have a race, and in each race, Nicholas can come in first, second, or third. The results of today's race are based only upon the results of yesterday's race. Let States 1, 2, and 3 represent that place that Nicholas comes in. The TPM is given below. 11/3 2/3 0 P= 0 1/2 1/2 L2/3 0 1/3 In order to determine an overall champion, the friends agree to play by a point system. In this system, the winner receives 2 points, the person in 2nd place receives 1 point, and the person in 3rd place receives no points. a. Given that Nicholas won today's race, what is the probability that he will win the next two races (tomorrow and the day after tomorrow)? b. Given that Nicholas won today's race, what is the probability that he will win the race the day after tomorrow? C. Suppose that on his very first race, Nicholas has a 50% chance of coming in 1st, a 25% chance of coming in 2nd, and a 25% chance of coming in 3rd. What is the probability that he will win the second race he runs? d. Given that Nicholas won the first race, how many total points do you expect him to have after the second day of racing? e. Find the steady-state probabilities for this system. f. In some race day in the distant future, how many points would you expect Nicholas to earn? g. Assuming that the competition lasts for N (a large number) of days, how many total points would you expect Nicholas to earn? h. If Nicholas came in third in the first race, what is the probability that he will win for the first time 3 days later? i. If Nicholas came in third in the first race, what is the mean number of days it will take him to come in first? j. If Nicholas came in first today, how many days on average will it take him to come in first again? k. If Nicholas came in third today, how many races is he expected to win out of the next 5?

Explanation / Answer

Let X0=1, X1=2 and X2=3 denote the state of the race then

Now from TPM

P12 =2/3   and      P23 =1/2

Hence, P(X1=2, X2=3/ X0=1)         =             2/3 *1/2

                                                =             1/3

                                                =             0.33                                                       

Now P2

P2=         1/3         2/3         0                              1/3         2/3         0

                0              1/2         1/2         *             0              1/2         1/2

                2/3         0              1/3                         2/3         0              1/3

                               

                1/9         10/18     1/3

=             1/3         1/4         5/12

                4/9         4/9         1/9                        

Therefore, (P2)13                     =             1/3

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