Paper 2, Section II, F

I was given a clockwork orange for my birthday. Initially, I place it at the centre of my dining table, which happens to be exactly 20 units long. One minute after I place it on the table it moves one unit towards the left end of the table or one unit towards the right, each with probability 1/2. It continues in this manner at one minute intervals, with the direction of each move being independent of what has gone before, until it reaches either end of the table where it promptly falls off. If it falls off the left end it will break my Ming vase. If it falls off the right end it will land in a bucket of sand leaving the vase intact.

(a) Derive the difference equation for the probability that the Ming vase will survive, in terms of the current distance $k$ from the orange to the left end, where $k=1, \ldots, 19$.

(b) Derive the corresponding difference equation for the expected time when the orange falls off the table.

(c) Write down the general formula for the solution of each of the difference equations from (a) and (b). [No proof is required.]

(d) Based on parts (a)-(c), calculate the probability that the Ming vase will survive if, instead of placing the orange at the centre of the table, I place it initially 3 units from the right end of the table. Calculate the expected time until the orange falls off.

(e) Suppose I place the orange 3 units from the left end of the table. Calculate the probability that the orange will fall off the right end before it reaches a distance 1 unit from the left end of the table.

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