EXERCISE #3:  PROBABILITY AND EARTHQUAKES

To complete Part 1 of this exercise, you will need to have access to twenty 10-sided die.  If you don't have any of these, some are available for use in Room D-320 of the Science Building.

Your objective in this die-rolling experiment is to learn how the probability of rolling a zero changes with the number of die rolled.  Complete the table below by rolling the number of die indicated in the left hand column (starting with one die and working up to 20) ten times (each roll is considered a "trial").  If at least one zero is rolled in a given trial, record a "1" in the appropriate trial column (do not record the number of zeroes, only whether at least one zero was rolled).  If no zeroes were rolled during a particular trial, leave that column blank.

Calculate the probability of rolling a zero for the number of die used by:   (1) adding the number of "1s" in  any given row, (2) dividing the sum by 10, and then (3) multiplying by 100 percent.

Example: Assume that when rolling three die at the same time, at least one zero was rolled in Trials 2, 5, 6,  and 10 (i.e., four times). The probability of rolling a zero with three die is then estimated to be 4 divided by 10 multiplied by 100%, or 40%. 

Plot the results of your experiment using the piece of graph paper provided with this assignment. On the vertical axis is the probability of rolling a zero (from last column in the table), and on the horizontal axis is  the number of die rolled (from Column 1 in the table).  Draw in a smooth curve that best fits the points on your graph.

1.  How does the shape of your curve compare with the theoretical curve based on statistics?

2.  According to your graph, how many die are needed to have a 50% chance of rolling a zero?

3.  According to your graph, how many die are needed to have a 90% chance of rolling a zero?

4.  The points you plotted to answer Question 1 probably do not fall exactly on a smooth curve.
     What change in the experiment would improve the fit of your data to a smooth curve?

PART 2:  Earthquakes and Cumulative Probability

To complete this part of the assignment, you should read pages 120-121 in your textbook.  You will also need to read the U.S.G.S. News Release that describes recent efforts to predict the next large earthquake at Parkfield, CA.  According to this 1998 News Release, Parkfield, CA has experienced six relatively large (M-6) earthquakes since record-keeping began in 1857.  The long-awaited 7th earthquake occurred in September 2004.  Use this information to answer the following questions.

1.  What are the six time spans separating the seven historic earthquakes at Parkfield?  Do these
     six time spans suggest that earthquakes occur here at random time intervals? 

2.  Estimate a recurrence interval  for M-6 earthquakes at Parkfield from the six time spans.

3.  Using the recurrence interval that you calculated for Question 2, calculate the one-year risk of a
      magnitude-6 earthquake occurring at Parkfield (assuming that earthquakes are random events).

4.  What is the cumulative risk of a magnitude-6 earthquake occurring during any 30-year period?

5.  The U.S. Geological Survey estimated the probability of a magnitude-6 earthquake occurring at
     Parkfield during the 30-year period between 1988 and 2018 to be greater than 90%.  How does
     this compare with your answer to Question 4?  How can you account for the difference?
 

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