Radioactivity: (ii) simulation of radioactive decay using dice

The idea

Radioactive decay is random and spontaneous, so a single nucleus's fate cannot be predicted, yet large samples show exponential decay. A dice simulation models this: each die represents a nucleus, and a chosen face (say a 6) means that nucleus has decayed. Roll all dice, remove those showing 6, and record the number remaining after each throw. The fraction surviving per throw is 5/6, so the count falls by a constant factor each round, producing an exponential curve. With many dice the decay is smooth; with few, random fluctuation is visible. This links the probabilistic nature of decay to the decay constant λ and the equation N = N₀e^(−λt).

What earns the marks

• Each die models one unstable nucleus, and a chosen face represents that nucleus decaying in that time interval.
• Removing dice that show the chosen face and counting survivors each round produces data that fall by a constant fraction per throw.
• The simulation shows decay is random for individual nuclei but predictable in the statistical behaviour of a large sample.
• Plotting the number remaining against throw number gives an exponential curve, analogous to N = N₀e^(−λt).
• Increasing the number of dice reduces the relative fluctuation, so the curve becomes smoother and closer to the theoretical exponential.

Common mistakes and how to put them right

• Thinking the dice 'remember' previous rolls.

  Put it right: Each throw is independent, so a surviving die has the same decay probability each round.

• Believing the number remaining falls by a fixed amount each throw.

  Put it right: It falls by a constant fraction, which is why the graph is exponential, not linear.

• Assuming a small number of dice will give a smooth exponential curve.

  Put it right: Random fluctuation is large, so more dice are needed for a clear trend.

• Confusing the decayed dice with the surviving dice when plotting.

  Put it right: The count that decays exponentially is the number remaining.

Exam tips

• State clearly what each die and each chosen face represent before describing the procedure.
• Use the simulation to explain randomness and the statistical nature of decay, not to claim individual nuclei have predictable lifetimes.
• When interpreting a graph, describe the constant ratio between successive counts rather than a constant difference.