**ERIC Number:**EJ1035557

**Record Type:**Journal

**Publication Date:**2014-Aug

**Pages:**5

**Abstractor:**ERIC

**Reference Count:**3

**ISBN:**N/A

**ISSN:**ISSN-0025-5769

Who's in the Lead?

Flores, Alfinio

Mathematics Teacher, v108 n1 p18-22 Aug 2014

Tossing a fair coin 1000 times can have an unexpected result. In the activities presented here, players keep track of the accumulated total for heads and tails after each toss, noting which player is in the lead or whether the players are tied. The winner is the player who was in the lead for the higher number of turns over the course of the game. Students are in for many surprises: The numbers of ties along the way is far lower than they expected; it is not unlikely that one player will be in the lead a very high percentage of the time (Peterson 1998, p. 16); and students see that two different measures--the relative frequency of heads (or tails) and the portion of times in the lead--can behave very differently. Students learn that the law of large numbers does not imply any regularity in the outcomes in the short term (such as frequent alternations of tails and heads or frequent ties). The law of large numbers states that as the number of trials increases, the experimental probability is more likely to be close to the theoretical probability. Thus, in the long run, when flipping coins, one may reasonably expect that the relative portion of heads (or tails) will be close to the theoretical probability. However, one common misconception is believing that for short sequences of random trials the empirical probability will also be close to the theoretical probability. This misconception is sometimes referred to as "the belief in the law of small numbers" (Tversky and Kahneman 1971). The activities presented in this article help students become aware of such misconceptions, make them explicit, and confront and correct them. The purpose of the activities is to help students develop a deeper understanding of what the law does and does not say. They add a layer of complexity by keeping track of outcomes in two ways--recording not only the outcome of the toss but also who is in the lead after each toss.

Descriptors: Mathematics Instruction, Learning Activities, Numbers, Mathematical Concepts, Probability, Misconceptions, Prediction, Experiments, Computation, Simulation, Educational Games, Calculators, High School Students, Secondary School Mathematics

National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191-1502. Tel: 800-235-7566; Tel: 703-620-3702; Fax: 703-476-2970; e-mail: orders@nctm.org; Web site: http://www.nctm.org/publications/

**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**High Schools; Secondary Education

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A