One coin proves nothing. A thousand coins prove a law.
Each toss is pure chaos: nobody on earth can call it. Yet toss a thousand and the running frequency of heads walks, stumbling, straight to $\tfrac12$. That walk is the law of large numbers — the only reason casinos, insurers, and physicists sleep at night.
The mint
tosses0
heads0
frequency—
distance to ½—
the running frequency, walking home to one half
The coin has no memoryfive heads in a row promise nothing about the sixth. The law works by drowning streaks, not by correcting them.
02
Two dice
Eleven outcomes, but not eleven chances — the middle is heavy.
Roll two dice and add. There is exactly one way to make $2$, and six ways to make $7$: the sums pile into a triangle. Marine dots are the theory, $\frac{6-|s-7|}{36}$; gold bars are your actual rolls. Watch the experiment climb into the theory’s shape.
The table
rolls0
last roll—
sevens seen—
theory says16.67%
gold bars: your rolls · marine dots: the triangle of theory
03
The tree
Two draws from one urn — multiply along the branch, add across the leaves.
An urn holds red and blue balls. Draw twice. The tree does all the bookkeeping: each branch carries its probability, each leaf multiplies its path, and any question — same colour? at least one red? — is just a sum of leaves. Switch to sans remise and watch the second row of branches shift.
The urn
red balls3
blue balls2
P(same colour)—
P(at least one red)—
multiply along a path · add the leaves you care about
04
The binomial
n tries, chance p each — the whole family of outcomes at one glance.
Repeat an experiment $n$ times with success chance $p$: the count of successes follows the binomial law, $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$. Slide $n$ and $p$ and watch the hill move (its top sits near $np$) and breathe (its width is $\sqrt{np(1-p)}$).
The dials
n tries12
p chance of success0.5
mean np—
spread σ—
most likely k—
P(that k)—
the guide line stands at np, the top of the hill
05
Galton’s board
Balls bounce left or right at every peg — and build the binomial by falling.
Drop a ball onto a wall of pegs: at each row it goes left or right, a coin toss made of wood. Where it lands is the number of rights it chose. Drop hundreds, and the bins draw the binomial hill by themselves — chaos, filing itself into order.
The board
rows of pegs10
balls dropped0
last ball landed in—
camellia: the last ball’s path · marine dots: the binomial it is building
06
The positive test
A 99% accurate test comes back positive. Your real risk may still be small.
A rare disease, an excellent test, a positive result — and yet. Among $1000$ people the sick are few, so even a small false-positive rate produces a crowd of healthy positives that can outnumber the true ones. The question that matters is $P(\text{malade}\mid\text{positif})$, and it is computed against the whole population, not against the test’s brochure.
The numbers
prévalence % of people sick1%
sensibilité % of sick detected99%
spécificité % of healthy cleared95%
—P(malade | positif)
true positives—
false positives—
missed sick—
cleared healthy—
1000 people · camellia full: sick & caught · camellia ring: sick & missed · vamp: healthy but positive
P(A|B) ≠ P(B|A)“99% of the sick test positive” and “99% of positives are sick” are different worlds. The second depends on how rare the disease is.
Épreuves the page checks by itself—
Six commissions against chance itself — and an endless examen. The page counts; luck does not.
07
Pitfalls
The six ways chance humiliates intuition.
No memoryafter five heads, the sixth toss is still 50/50. Streaks are drowned by volume, never repaid.
Independent ≠ incompatibleindependent events can happen together — that is the point. Incompatible ones never do.
P(A∩B) ≤ P(A)adding a detail can only make a story less probable, however vivid it sounds.
Inverted conditionals$P(\text{malade}\mid\text{positif})$ is not $P(\text{positif}\mid\text{malade})$. Chapter VI is the antidote.
Small samples lieten tosses wander anywhere. The law of large numbers only signs contracts in bulk.
A fair game still ruins youzero expected value says nothing about the variance that empties a pocket first.