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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- Explain how you would simulate the drunkard's walk, making efficient use of one-digit random numbers.
- 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.
- 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 \(\frac { 1 } { 3 }\).
- Explain how you would now simulate this situation.
- 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.)