PART II: Probability and Statistics

When you toss a single coin, it will come up either "heads" (H) or "tails" (T). As long as it isn't unevenly weighted, it should come up H half of the time and T the other half. But does that really happen in practice? For the following exercise you can toss your own coin, or you can make use of  the Coin Tosser JAVA Applet farther down this page. Flip the coin once; it will come up either H or T. Now flip it again, it will come up either H or T also. For the two flips there are four possible sequences of results:

We say that the chances that a single coin toss turns up H are 50:50, 50% of the time H and 50% of the time T. What this really means is that if you were to flip a coin one million times, half of the time it would come up H and half of the time it would come up T. But where the number of times you toss the coin is small (i.e., the number of times you "sample the data"), the number of times it turns up H isn't quite 50% of the total number of tosses. The probability that it is H will be somewhat more or less than one-half.

The difference in probability between what you measure and what you predicted by probability theory depends on the number of measurements you make. If you flip the coin N times, how large is the deviation from one-half that you measure?

The coin tosser is immediately below this text. By pushing the buttons, you can toss coins one, ten, or one hundred at a time. The number of heads and tails and the percentage of the total that are heads and tails are displayed.

Using the coin tosser, convince yourself that the percentage of tosses that turn up H will eventually approach 50% if you toss the coin enough times, but that it can be quite different from 50% when the number of tosses is small. The coin tosser works by randomly assigning H or T each time you tell it to toss the coin. We are interested in knowing how different from 50% the probability is that the coin will turn up H 50% of the time if we toss the coin N times, and how the deviation changes as N increases. Coin tossing is easy to do, but other measurements are much more time consuming and laborious. We don't want to make too many measurements, but we want to make enough that we can be sure to get the right answer (almost).

Probability theory tells us that if you flip the coin N times, the deviation from 50% that you would find is about in N tosses. For 10²⁰ particles (the number of atoms in a typical sample of radioactive material), a deviation of 10¹⁰ is totally negligible. With the numbers that we will obtain in counting radioactive decays in the second half of this lab, however, such fluctuations as expected purely by the laws of probability are important. All of your measurements will thus be of the form , and you should remember that your measurements are not exact but have some amount of uncertainty because of their limitations.

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