Question

A frog is sitting exactly in the middle of a board that is five feet long. Every ten seconds he jumps one foot either left or right, at random. You want to usesimulation to estimate the probability that he jumps off the board in sixty seconds or less.
a) Describe how you would use the digits of a random number table to simulate this chance process and estimate the probability.

Answer 1

 A frog sitting in the center of a board of length 5 feet is 2.5 feet from any end, Thus it needs a grid total of 3 one-foot jumps to the left or to the right in order to jump off of the board.

 If the frog makes its initial jump at time zero, next you need to think of a variety of ways that the frog might create seven jumps (i.e., at 0,10,20, 30,40,50, and 60 seconds).

It is simple to see that the frog will jump off of the board in 60 seconds or less in any of the following situations.

1. The frog's first 3 jumps are all in a similar direction

2. The frog's first 5 jumps are all in the same direction except for one jump in the reverse direction.

3. The frog's first 7 jumps are all in a similar direction except for 2 jumps in the reverse direction

Obviously, if the situation described by (1) is obtained, then you do not need to assess the situation (2). Likewise, if the state of the situation (2) is obtained, then you do not need to appraise the situation (3)

- This is because once the frog has jumped off of the board, the trial is a "success" and there is no reason to deem the frog's jumps any further.

If the frog has not jumped off of the board in its primary 7 jumps, then the sixty-second-time limit has been reached and the trial is a "failure."

To simulate this procedure using digits of a random number table, you will need to decide on some appropriate mapping between the sequence of digits in the table and the individual jumps of the frog.

For example, if the frog is evenly likely to jump left as it is to jump right, then you could treat any digit in {0,1,2,3,4} as meaning "frog jumps left," and any digit in {5,6,7,8,9} as meaning " frog jumps right.

For each replicated trial, you would take the next 7 digits from the table and estimate them to decide if one of the "success" circumstances describe above is obtained.

- Record the result of the trial, and then do it again and again, each time with another seven digits from the table.

- The ratio of the number of successes to the total number of trials can be used as an approximation of the probability that the frog jumps off the board in sixty seconds or less. Though, that the query does NOT state that the frog is similarly likely to jump left as it is to jump right. In choosing your mapping, you need to identify the probability that the frog jumps left or right on each prospect. If the frog is twice as likely to jump right as it is to jump left, then the probability of jumping left is 1/3 and the probability of jumping right is 2/3

In these circumstances, you could map digits in {0,1,2} to denote "frog jumps left" and digits {3,4,5,6,7,8} to mean " frog jumps right," and any time you get the digit 9 you ignore it and use the subsequent digit in the table that is less than 9.

If instead the probability that the frog jumps left is 0.57, then you could take numbers from the table in pairs to appraise each jump. As a result, if the paired digits are in {00,01,02, ..., 56},this could be treated as meaning "frog jumps left," and all other pairs in {57,58,59, .. ., 99} meaning "frog jumps right.

In this latter example, you would be taking up to seven pairs of digits from the table, indifferent to taking seven single digits as described in the previous paragraph. The exacting mapping you decide for the digits from the table will frequently depend on the known probability that the frog jumps left vs. right, as well as how exactly you want to represent that probability, and is usually not unique