2025

Experiment: Flip a fair coins

Flip1 X (Heads) P(X)
H 1 0.5
T 0 0.5

X = sum of heads in 1 fair coin flip

X P(X)
0 0.5
1 0.5

Experiment: Flip two fair coins

Flip1 Flip2 X (Heads) P(X)
H H 2 0.25
H T 1 0.25
T H 1 0.25
T T 0 0.25

X = sum of heads in 2 fair coin flips

X P(X)
0 0.25
1 0.50
2 0.25

Experiment: Flip three fair coins

Flip1 Flip2 Flip3 X (Heads) P(X)
H H H 3 0.125
H H T 2 0.125
H T H 2 0.125
H T T 1 0.125
T H H 2 0.125
T H T 1 0.125
T T H 1 0.125
T T T 0 0.125

X = sum of heads in 3 fair coin flips

X P(X)
0 0.125
1 0.375
2 0.375
3 0.125

Experiment: Flip four fair coins

Flip1 Flip2 Flip3 Flip4 X (Heads) P(X)
H H H H 4 0.0625
H H H T 3 0.0625
H H T H 3 0.0625
… … … … … …
H T T T 1 0.0625
T T T T 0 0.0625

X = sum of heads in 4 fair coin flips (unique outcomes)

X P(X)
0 0.0625
1 0.25
2 0.375
3 0.25
4 0.0625

X = sum of heads in 5 fair coin flips

Flip1 Flip2 Flip3 Flip4 Flip5 X (Heads) P(X)
(32 combinations omitted for brevity)

X = sum of heads in 5 fair coin flips (unique outcomes)

X P(X)
0 0.03125
1 0.15625
2 0.31250
3 0.31250
4 0.15625
5 0.03125

X = sum of heads in 100 fair coin flips

Flip1 Flip2 … Flip100 X (Heads) P(X)
(2^100 combinations omitted for obvious reasons)

X = sum of heads in 100 fair coin flips (unique outcomes)

X P(X)
0 to 100 Approximates a normal distribution centered at 50

Summary: What We Observed

  • We defined \(X\) as the sum of heads in a sequence of fair coin flips.

  • For small numbers of flips (e.g., 1–3), the distribution of \(X\) was clearly discrete and not very bell-shaped.

  • As the number of flips increased, the distribution of \(X\) became increasingly bell-shaped and looked increasingly continuous.

  • This gradual emergence of a bell shape — even though individual outcomes are binary — is a demonstration of the Central Limit Theorem.