2025

Notation

We now define the random variable \(\bar{X}\) as the mean number of heads from flipping \(n\) fair coins.

This is computed by dividing the number of heads by \(n\). The possible values of \(\bar{X}\) will lie between 0 and 1.

Experiment: Flip 1 fair coin

Flip1 \(\bar{X}\) \(P(\bar{X})\)
H 1.00 0.5
T 0.00 0.5

\(\bar{X}\) = mean number of heads in 1 fair coin flip

\(\bar{X}\) \(P(\bar{X})\)
0.00 0.5
1.00 0.5

Experiment: Flip 2 fair coins

Flip1 Flip2 \(\bar{X}\) \(P(\bar{X})\)
H H 1.00 0.25
H T 0.50 0.25
T H 0.50 0.25
T T 0.00 0.25

\(\bar{X}\) = mean number of heads in 2 fair coin flips

\(\bar{X}\) \(P(\bar{X})\)
0.00 0.25
0.50 0.50
1.00 0.25

Experiment: Flip 3 fair coins

Flip1 Flip2 Flip3 \(\bar{X}\) \(P(\bar{X})\)
H H H 1.00 0.125
H H T 0.67 0.125
H T H 0.67 0.125
H T T 0.33 0.125
T H H 0.67 0.125
T H T 0.33 0.125
T T H 0.33 0.125
T T T 0.00 0.125

\(\bar{X}\) = mean number of heads in 3 fair coin flips

\(\bar{X}\) \(P(\bar{X})\)
0.00 0.125
0.33 0.375
0.67 0.375
1.00 0.125

Experiment: Flip 100 fair coins

Flip1 Flip2 … Flip100 \(\bar{X}\) \(P(\bar{X})\)
(2^100 combinations omitted)

\(\bar{X}\) = mean number of heads in 100 fair coin flips

\(\bar{X}\) \(P(\bar{X})\)
0.00 to 1.00 Approximates a normal distribution centered at 0.50

Summary: What We Observed

  • We defined \(\bar{X}\) as the mean number of heads in a sequence of fair coin flips.

  • For small \(n\), \(\bar{X}\) had only a few possible values and did not have a bell-shaped distribution.

  • As \(n\) increased, the distribution of \(\bar{X}\) 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.