OBJECTIVE To experience how random events can lead to exponential decay models.

Radioactive elements decay when their atoms spontaneously emit radiation and change into smaller, stable atoms. But if atoms decay randomly, how is it possible to find a function that models their behavior? We’ll try to answer this question by experimenting with coins and dice.

Imagine tossing a coin $100$ times (or, equivalently, tossing $100$ coins all at once). How often would you expect to get $100$ heads? Or $99$ heads? A much more likely outcome is that the number of heads and tails will be about the same (because the “rate” at which heads appears is $\text{50%}$). So although the outcome of a single toss of a coin is totally unpredictable, when we toss many coins, the number of heads is fairly predictable. Of course, there are a lot more than $100$ atoms in even the tiniest sample of a radioactive substance, so the outcome of the random decay of atoms in the sample is quite predictable. Let’s simulate the random decay of radioactive atoms using coin tosses and rolls of dice.

I. Modeling Radioactive Decay with Coins

In this first experiment we toss pennies to simulate the decay of atoms in a radioactive substance.

You will need:

• $40$ pennies
• A cup or jar
• A table

Procedure:

1. Put the pennies in a cup and shake them well, then toss them on a table. The pennies that show tails are considered "decayed," and those that show heads are still "radioactive."
2. Discard the decayed pennies and collect the radioactive ones. Record the number of radioactive pennies remaining (in the table below).
3. Repeat Steps 1 and 2 with the remaining radioactive pennies until all the pennies have decayed.

   Toss number $x$	Pennies remaining $f(x)$	Ratio $\frac{f(x+1)}{f(x)}$
   $0$	$40$	—
   $1$
   $2$
   $3$
   $4$
   $5$
   ⁝

Analysis:

4. Is an exponential model appropriate for the data you obtained? To decide, complete the "Ratio" column in the table to determine whether there is a reasonably constant "decay factor."
5. 1. Use the E x p R e g command on a graphing calculator to find the exponential curve $Y = a \cdot b^x$ that best fits the data: $$Y = \color{red}{\fbox{   }} \cdot \color{red}{\fbox{   }}^x$$
   2. What is the decay factor for the model you found in part (a)? Is the decay factor what we should expect when tossing pennies?

II. Modeling Radioactive Decay with Dice

In this experiment we roll dice to simulate the decay of atoms in a radioactive substance.

You will need:

• $24$ dice
• A cup or jar
• A table

Procedure:

This is basically the same experiment as in Part I, except with dice.

1. Put the dice in a cup and shake them well, then roll them on a table. The dice that show a one or a six are considered "decayed," and the others are still "radioactive."
2. Discard the decayed dice and collect the radioactive ones. Record the number of radioactive dice remaining (in the table below).
3. Repeat Steps 1 and 2 with the remaining radioactive dice until all the dice have decayed.

   Toss number $x$	Pennies remaining $f(x)$	Ratio $\frac{f(x+1)}{f(x)}$
   $0$	$24$	—
   $1$
   $2$
   $3$
   $4$
   $5$
   ⁝

Analysis:

4. Is an exponential model appropriate for the data you obtained? To decide, complete the "Ratio" column in the table to determine whether there is a reasonably constant "decay factor."
5. 1. Use the E x p R e g command on a graphing calculator to find the exponential curve $Y = a \cdot b^x$ that best fits the data: $$Y = \color{red}{\fbox{   }} \cdot \color{red}{\fbox{   }}^x$$
   2. What is the decay factor for the model you found in part (a)? Is the decay factor what we should expect when rolling dice?