Buffon's Needle

What is Buffon's Needle?

This activity allows the user to run a simulation of dropping a needle on a lined sheet of paper and determining the probability of the needle crossing one of the lines. The remarkable result is that the probability is directly related to the value of Pi. By using calculus, one can prove (as people first discovered in the eighteenth century) that the probability of a hit is 2/Pi, where Pi is defined as the ratio of a circle's circumference to its diameter. To calculate Pi from the needle drops, the computer program finds experimental probability:

Experimental Probability = (# of hits)/(total # of drops)

On the other hand, we have (approximately):

Theoretical Probability = 2/Pi

We can use this fact to conclude that:

2(total # of drops)/(# of hits)= Pi (approximately)

How Do I Use This Activity?

This activity allows the user to run a simulation of dropping a needle on a lined sheet of paper and determining the probability of the needle crossing one of the lines.

Controls and Output

  • In the graphic at the top of the applet, the red line represents the needle. The current estimate of pi is printed in black text. If the text reads "Run More Trials" instead of "PI = (some number)," then there have not been enough needle drops to estimate pi, and you need to run more trials.
  • To drop the needle on the sheet just once, press the New Pin button.
  • The more times you drop the needle, the more accurate the estimate of pi will be. To run many trials at once, type the number of trials you want to run in the box labeled Trials =. Then press the Run Trials button.
  • If you want the results from every single trial to be added together and used in calculating pi, check the box marked Keep running total. If you do not check this box, every time you click New Pin or Run Trials all the data will be reset.

Description

This activity allows the user to run a simulation of dropping a needle on a lined sheet of paper and determining the probability of the needle crossing one of the lines. This activity would work well in groups of two for about ten to fifteen minutes if you use the exploration questions and five to ten minutes otherwise.

Place in Mathematics Curriculum

This activity can be used to:

  • introduce the notions of chance and probability
  • show the difference between experimental and theoretical probability
  • motivate the connection between probability and geometry

Standards Addressed

Grade 6

  • Statistics and Probability

    • The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 7

  • Statistics and Probability

    • The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 8

  • Statistics and Probability

    • The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 9

  • Statistics and Probability

    • The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 10

  • Statistics and Probability

    • The student demonstrates a conceptual understanding of probability and counting techniques.

Seventh Grade

  • Statistics and Probability

    • Investigate chance processes and develop, use, and evaluate probability models.

Number and Quantity

  • Quantities

    • Reason quantitatively and use units to solve problems.

Statistics and Probability

  • Conditional Probability and the Rules of Probability

    • Understand independence and conditional probability and use them to interpret data
    • Use the rules of probability to compute probabilities of compound events in a uniform probability model

  • Making Inferences and Justifying Conclusions

    • Understand and evaluate random processes underlying statistical experiments
    • Make inferences and justify conclusions from sample surveys, experiments, and observational studies

  • Using Probability to Make Decisions

    • Calculate expected values and use them to solve problems
    • Use probability to evaluate outcomes of decisions

Grades 9-12

  • Data Analysis and Probability

    • Develop and evaluate inferences and predictions that are based on data
    • Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
    • Understand and apply basic concepts of probability

Advanced Functions and Modeling

  • Data Analysis and Probability

    • Competency Goal 1: The learner will analyze data and apply probability concepts to solve problems.

Grade 6

  • Probability and Statistics

    • 9. The student uses experimental and theoretical probability to make predictions.

Grade 8

  • Probability and Statistics

    • 11. The student applies concepts of theoretical and experimental probability to make predictions.

7th Grade

  • Probability and Statistics

    • 7.14 The student will investigate and describe the difference between the probability of an event found through simulation versus the theoretical probability of that same event.

Be Prepared to

  • give implicit directions on what they are to do. For example "Today we are going to run trials onthis applet and see if a pattern appears..."
  • answer the question "Why is the probability of the needle landing on a line related to Pi?"
  • show how Pi was created.