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A probability can shrink toward zero while its event still occurs infinitely many times. The missing question is whether the probabilities are summable and how the events depend on one another. The Borel–Cantelli lemmas turn that question into a disciplined test: define the infinitely-often event, inspect the probability series, and state exactly where independence is or is not required.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. Define “infinitely often” before calculating
For events A1, A2, …, the event “An occurs infinitely often” contains outcomes that belong to infinitely many of the An. It is written An i.o. or lim sup An.
Set notation makes the timing explicit: lim sup An is the intersection, over every starting index m, of the union of all An with n at least m. No matter how far along the sequence we start, another occurrence remains.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Use the first lemma without inventing independence
If the series ΣP(An) converges, the first Borel–Cantelli lemma gives P(An i.o.)=0. With probability one, only finitely many of the events occur.
This direction does not require independence. A union bound controls the probability of any occurrence after time m by the tail sum Σn≥m P(An), and that tail approaches zero when the series converges.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Work a summable-event register
Suppose the probability of a quality-control false alarm at inspection n is 1/n². The probabilities decrease and their infinite sum converges.
| Inspection n | P(An) | Selected cumulative tail clue | Interpretation |
|---|---|---|---|
| 1 | 1 | Large at the start | Early alarms can occur |
| 2 | 1/4 | Tail still visible | No claim about a short run |
| 10 | 1/100 | Small individual chance | Small is not the proof |
| n→∞ | 1/n² | Σ1/n² converges | Only finitely many alarms almost surely |
The lemma does not say how many alarms occur or supply a deterministic last alarm time. “Almost surely finitely many” is a probability-one statement about the infinite sequence.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Apply the second lemma with its condition visible
If the events An are independent and ΣP(An) diverges, the second Borel–Cantelli lemma gives P(An i.o.)=1. Infinitely many events occur almost surely.
For independent events with P(An)=1/n, the harmonic series diverges. Although each event becomes individually rare, the sequence supplies enough cumulative opportunity for infinitely many occurrences with probability one.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. See why divergence alone is insufficient
Let U be uniform on (0,1] and define An={U≤1/n}. Then P(An)=1/n and the probability series diverges, but the events are nested downward.
| Feature | Independent 1/n events | Nested events {U≤1/n} |
|---|---|---|
| Marginal probability | 1/n | 1/n |
| Series | Diverges | Diverges |
| Dependence | Independent | Strongly dependent and decreasing |
| P(An i.o.) | 1 | 0 |
| Lesson | Second lemma applies | Divergence alone does not decide |
For the nested sequence, occurring infinitely often means U must be no greater than 1/n for arbitrarily large n, which has probability zero. Dependence changes the conclusion.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Distinguish rare events from finite expected counts
The sum ΣP(An) is also the expected total count of occurrences when the count is written as a sum of indicators, even without independence. A finite expected count rules out an infinite count with positive probability.
A divergent expected count does not by itself force infinitely many occurrences. The nested example concentrates many nominal opportunities on the same underlying outcome.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: threshold exceedances
A monitoring system raises event An when an error exceeds a threshold at time n. If valid bounds give P(An)≤C/n², comparison with a convergent series shows that exceedances occur only finitely often almost surely.
If the bound is instead C/n, divergence prevents the first lemma from settling the question. Independence, weaker dependence conditions or a different argument must be established rather than assumed.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect dependence and convergence
Use reinforced urns and dependent trials to see how equal marginal probabilities can hide different histories. Read probability convergence for the wider distinction among almost-sure, probability and distributional limits.
For finite stopping and resource caps, continue with first success, retry limits and unfinished runs.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
P(An)→0 does not imply that only finitely many An occur. A decreasing individual chance can still have a divergent total opportunity.
The first lemma never needs independence. The common elementary form of the second does. Neither lemma says that probability-one behaviour happens for every possible outcome.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence and checking routine
Write the event sequence, the limsup event and the relevant probability series. Test convergence before discussing dependence.
For the divergent case, document the joint structure. If independence is unavailable, state that Borel–Cantelli II cannot be invoked in its elementary form and identify the alternative dependence result or model evidence being used.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Does probability zero mean impossible?
Not in every continuous or infinite model. A probability-zero set can contain outcomes; the lemma makes an almost-sure statement.
Why does the first lemma not need independence?
A union bound and a vanishing tail sum are enough to control any occurrence after a late index.
Can a divergent series still give only finitely many events?
Yes. Strong dependence can do so, as the nested uniform-variable example shows.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
MIT OpenCourseWare’s Fundamentals of Probability Lecture 3 defines infinitely-often events and develops the Borel–Cantelli lemma. MIT’s Borel–Cantelli seminar paper states both directions and their independence conditions. The inspection and nested-event cases are original teaching examples.
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