Multi-stage probability simulation

6 An apple tree has 6 apples left on it. Each day each remaining apple has a probability of \(\frac { 1 } { 3 }\) of falling off the tree during the day.

1. Give a rule for using one-digit random numbers to simulate whether or not a particular apple falls off the tree during a given day.
2. Use the random digits given in your answer book to simulate how many apples fall off the tree during day 1 . Give the total number of apples that fall during day 1 .
3. Continue your simulation from the end of day 1 , which you simulated in part (ii), for successive days until there are no apples left on the tree. Use the same list of random digits, continuing from where you left off in part (ii). During which day does the last apple fall from the tree? Now suppose that at the start of each day the gardener picks one apple from the tree and eats it.
4. Repeat your simulation with the gardener picking the lowest numbered apple remaining on the tree at the start of each day. Give the day during which the last apple falls or is picked. Use the same string of random digits, a copy of which is provided for your use in this part of the question.
5. How could your results be made more reliable?

5 The diagram shows the progress of a drunkard towards his home on one particular night. For every step which he takes towards his home, he staggers one step diagonally to his left or one step diagonally to his right, randomly and with equal probability. There is a canal three steps to the right of his starting point, and no constraint to the left. On this particular occasion he falls into the canal after 5 steps.
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1. Explain how you would simulate the drunkard's walk, making efficient use of one-digit random numbers.
2. Using the random digits in the Printed Answer Book simulate the drunkard's walk and show his progress on the grid. Stop your simulation either when he falls into the canal or when he has staggered 6 steps, whichever happens first.
3. How could you estimate the probability of him falling into the canal within 6 steps? On another occasion the drunkard sets off carrying a briefcase in his right hand. This changes the probabilities of him staggering to the right to \(\frac { 2 } { 3 }\), and to the left to \(\frac { 1 } { 3 }\).
4. Explain how you would now simulate this situation.
5. Simulate the drunkard's walk (with briefcase) 10 times, and hence estimate the probability of him falling into the canal within 6 steps. (In your simulations you are not required to show his progress on a grid. You only need to record his steps to the right or left.)

5 Each morning I reach into my box of tea bags and, without looking, randomly choose a bag. The bags are manufactured in pairs, which can be separated along a perforated line. So when I choose a bag it might be attached to another, in which case I have to separate them and return the other bag to the box. Alternatively, it might be a single bag, having been separated on an earlier day. I only use one tea bag per day, and the box always gets thoroughly shaken during the day as things are moved around in the kitchen. You are to simulate this process, starting with 5 double bags and 0 single bags in the box. You are to use single-digit random numbers in your simulation.

1. On day 2 there will be 4 double bags and 1 single bag in the box, 9 bags in total. Give a rule for simulating whether I choose a single bag or a double bag, assuming that I am equally likely to choose any of the 9 bags. Use single-digit random numbers in your simulation rule.
2. On day 3 there will either be 4 double bags or 3 double bags and 2 single bags in the box. Give a rule for simulating what sort of bag I choose in the second of these cases. Use single-digit random numbers in your simulation rule.
3. Using the random digits in your answer book, simulate what happens on days 2,3 and 4 , briefly explaining your simulations. Give an estimate of the probability that I choose a single bag on day 5 .
4. Using the random digits in your answer book, carry out 4 more simulations and record the results.
5. Using your 5 simulations, estimate the probability that I choose a single bag on day 5 .
   [0pt] [Question 6 is printed overleaf.]

6 Answer parts (ii)(A) and (iii)(B) of this question on the insert provided. A particular component of a machine sometimes fails. The probability of failure depends on the age of the component, as shown in Table 6. \section*{Table 6} You are to simulate six years of machine operation to estimate the probability of the component failing during that time. This will involve you using six 2-digit random numbers, one for each year.

1. Give a rule for using a 2-digit random number to simulate failure of the component in its first year of life. Similarly give rules for simulating failure during each of years 2 to 6 .
2. (A) Use your rules, together with the random numbers given in the insert, to complete the simulation table in the insert. This simulates 10 repetitions of six years operation of the machine. Start in the first column working down cell-by-cell. In each cell enter a tick if there is no simulated failure and a cross if there is a simulated failure. Stop and move on to the next column if a failure occurs.
   (B) Use your results to estimate the probability of a failure occurring. It is suggested that any component that has not failed during the first three years of its life should automatically be replaced.
3. (A) Describe how to simulate the operation of this policy.
   (B) Use the table in the insert to simulate 10 repetitions of the application of this policy. Re-use the same random numbers that are given in the insert.
   (C) Use your results to estimate the probability of a failure occurring.
4. How might the reliability of your estimates in parts (ii) and (iii) be improved?