Gaussian (Normal) Distribution
For independent coin flips, the distribution of the suitably scaled total can approach a Gaussian bell curve as the number of flips grows. Smaller stakes can hold variance constant while the number of flips increases.
THE IDEA
FIGURE 9: NUMBERS OF WINS TOSSED
The source describes a bell-shaped distribution emerging from repeated sets of forty tosses. The figure itself is not supplied.
We are not yet fully in a Gaussian bell curve, but we are getting dangerously close. This is still proto-Gaussian, but you can see the gist. (Actually, you will never encounter a Gaussian in its purity since it is a Platonic form—you just get closer but cannot attain it.) The source describes Figure 9 as showing this emerging bell shape.
How do we get even closer to the perfect Gaussian bell curve? By refining the flipping process. We can either flip 40 times for $1 a flip or 4,000 times for ten cents a flip, and add up the results. For independent fair flips, the variance of the total is the same in both situations. The equivalence in the two sets of flips has a little nonintuitive hitch. We multiplied the number of bets by 100, but divided the bet size by 10—don’t look for a reason now, just assume that they are “equivalent.” The variance is equal, but the range and tail probabilities differ. The source next refers to winning or losing 400 times in a row. The odds are about one in 1 with 120 zeroes after it, that is, one in 1,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000 times.
Continue the process for a while. We go from 40 tosses for $1 each to 4,000 tosses for 10 cents, to 400,000 tosses for 1 cent, getting close and closer to a Gaussian. The source describes Figure 10 as spreading results between −40 and 40, or eighty plot points, and a subsequent version as having 8,000 points. Those figures are absent. The passage also shifts between the total number of flips and the length of a run, so the quoted extreme-run probability should not be read as the chance of the maximum loss across all flips.
When to use it
Modeling measurement errors in scientific experiments.
Financial risk assessment: testing whether a normal approximation is suitable and examining dependence and tail losses before relying on it.
Quality control in manufacturing: monitoring product dimensions that follow a bell curve.
Educational testing: analyzing test scores that often approximate a normal distribution.
FROM THEORY TO EVERYDAY LIFE
In practice
Coin Toss Approximation
Flipping a fair coin multiple times produces a distribution of wins that approaches the Gaussian bell curve. For example, 40 tosses of $1 each gives a proto-Gaussian shape, while 4,000 tosses at 10 cents each produces a closer approximation, and 400,000 tosses at 1 cent each gets even closer.
A MOMENT TO REFLECT
Put it to work.
How would the distribution change if the coin was biased (e.g., 60% heads)?
What real-world phenomena follow a Gaussian distribution, and why might some not?
Why is it impossible to observe a perfect Gaussian in practice?
CONTINUE THE CONVERSATION
Source texts
- The Black Swan The Impact of the Highly Improbable With a new section On Robustness and Fragility (Nassim Nicholas Taleb)