The Dice Rolling Simulation

Each die in this experiment represents a radioactive nucleus. On every “roll” (i.e., time step), each die has a fixed probability ( p = ) of decaying (being removed). We begin with 80 dice.

This setup mimics radioactive decay, where: - The probability of a single atom decaying per unit time is constant. - The number of undecayed atoms decreases over time.

# Required libraries
library(ggplot2)

# Parameters
P0 <- 80
n_rolls <- 50
p <- 1 / 6

# Simulation
pop_avg <- numeric(n_rolls + 1)
set.seed(NULL)

P <- P0
pop_avg[1] <- P0
k <- 1
while (k <= n_rolls && P > 0) {
  r <- rbinom(1, P, p)
  P <- P - r
  pop_avg[k + 1] <- P
  k <- k + 1
}

# Truncate data when population hits zero
final_index <- which(pop_avg == 0)[1]
if (is.na(final_index)) {
  final_index <- length(pop_avg)
}
pop_avg <- pop_avg[1:final_index]
time <- 0:(length(pop_avg) - 1)

# Exponential decay model
lambda <- -log(1 - p)
exp_model <- P0 * exp(-lambda * time)

# Geometric approximation
geometric_expansion <- function(t, p, P0, n_terms = 15) {
  sapply(t, function(tt) {
    sum(sapply(0:(n_terms - 1), function(k) {
      coef <- (-1)^k * choose(tt, k)
      coef * p^k
    })) * P0
  })
}
geom_approx <- geometric_expansion(time, p, P0, n_terms = 15)

# Half-life
t_half <- log(2) / lambda

# Combine data
df <- data.frame(
  Roll = time,
  Experimental = pop_avg,
  Exponential = exp_model,
  Geometric = geom_approx
)

# Plot
ggplot(df, aes(x = Roll)) +
  geom_line(aes(y = Experimental, color = "Experimental (Simulated)")) +
  geom_line(aes(y = Exponential, color = "Exponential Decay (Continuous)")) +
  geom_line(aes(y = Geometric, color = "Geometric Expansion (15 terms)"), linetype = "dashed") +
  geom_vline(xintercept = t_half, linetype = "dotted", color = "purple", linewidth = 1) +
  geom_hline(yintercept = P0 / 2, linetype = "dotted", color = "purple") +
  annotate("text", x = t_half + 1, y = P0 / 2 + 3, label = "Half-life", color = "purple") +
  labs(
    title = "Radioactive Decay with Half-life Marker",
    x = "Roll Number", y = "Dice Remaining",
    color = "Legend"
  ) +
  theme_minimal()

What’s Plotted?

The graph shows three curves:

1. Experimental (Simulated) Line

• This line shows the actual number of dice remaining after each roll in a single experiment.
• It’s jagged because it’s random: on each roll, a different number of dice decay.

2. Exponential Decay Line (Continuous Model)

• This is the smooth curve you’d get if decay happened continuously (instead of once per roll).
• It’s based on the exponential decay formula:
  [ N(t) = N_0 e^{-t}, = -(1 - p) ]
• This is often taught as the theoretical decay law in physics.

🔍 This curve smooths out randomness and assumes infinitesimally small time steps — great for math, but not realistic for real-world random events.

3. Geometric Expansion Line (Dashed)

• This is a series approximation of the geometric decay:
  [ N(t) = N_0 (1 - p)^t ]
• The series is built using the binomial expansion of ( (1 - p)^t ), and you’ve included 15 terms for good accuracy even at early times.
• This curve lies right on top of the exponential line for small ( t ), showing they’re nearly equivalent at first.

What the Graph Shows You

• Agreement at Early Times: All three lines start at the same place and follow a similar shape, showing how well the continuous exponential model approximates the actual discrete decay early on.
• Divergence Through Randomness: The experimental line starts to diverge due to the random nature of decay (especially in single trials).
• Geometric vs Exponential: The geometric expansion sits between the raw simulation and the smooth exponential, acting as a bridge between theory and simulation.

Physics Context

This graph is a perfect classroom tool to: - Introduce exponential decay and how it’s derived. - Highlight the assumptions behind the exponential model (e.g., continuous time). - Show how probabilistic events lead to deterministic laws when averaged or modeled. - Provide a critical look at models — the “perfect curve” doesn’t represent any single real-world trial, but rather an average over many.