What does it mean that the probability of heads, when you toss an unbiased coin, is fifty percent?

Students work in small groups. They use the TI-83 or TI-84 calculator to simulate 9, 99, and 999 tosses of a coin. They gather several samples and record the number of (simulated) heads, and the deviation of their frequencies from 50%. They formulate and discuss their observations.

Example of a program:

(In this program percentage is rounded to 1%)

We assign an outcome of 1 to mean heads, and an outcome of 0 to mean tails. We count the number of ones (heads) in each sample.

Set MODE to Float 0. To get samples of size N = 9, 99, 999, first store 9 or 99 or 999 in N, and then, for each N, run the following sequences of keystrokes:

sum(randInt(0,1,

N))→A:100A/N-50

→B:{A,B} repeat ENTER 8 times to get 8 samples of each size.

Example of a simulation (deviation from 50% is in parentheses):

N(sample size): | 9 | 99 | 999 |

Sample number | No. of heads | No. of heads | No. of heads |

1 | 7(28%) | 41(-9%) | 478(-2%) |

2 | 3(-17%) | 48(-2%) | 503(0%) |

3 | 3(-17%) | 57(-2%) | 512(1%) |

4 | 5(6%) | 47(-3%) | 486(-1%) |

5 | 3(-17%) | 51(2%) | 515(2%) |

6 | 5(6%) | 52(3%) | 531(3%) |

7 | 3(-17%) | 40(-10%) | 502(0%) |

8 | 4(-6%) | 50(1%) | 499(-0%) |

Some points for discussion.

(1) In none of the cases is the frequency exactly 50%, because the sizes of the samples were odd numbers.

(2) Even when the sample is large, a big deviation from 50% is possible, but it is very unlikely.

(3) When we say that "the probability of heads is 50%", we mean that: When a sample is large, the chance that the frequency is very different from 50% is small.

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