Infinite Sequences of Bernoulli Trials

Definition

A sequence of random variables

$$X_1, X_2, X_3, \dots$$

is called an infinite sequence of Bernoulli trials if:

• Each $X_i$ takes only two values, usually $1$ and $0$
• $1$ = success
• $0$ = failure
• The trials are independent
• The probability of success is the same for every trial: $$P(X_i = 1) = p, \quad P(X_i = 0) = 1-p$$ where $0 \le p \le 1$

If this holds for all $i = 1, 2, 3, \dots$, then the process is an infinite Bernoulli sequence.

A common notation is:

$$X_i \sim \text{Bernoulli}(p)$$

for every $i$, with the collection being independent across all trials.

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Main Content

1. Bernoulli Trial and Bernoulli Sequence

• A single Bernoulli trial is one experiment with exactly two outcomes. For example, a coin toss can be modeled as:
• Head = success
• Tail = failure
• An infinite Bernoulli sequence is the repeated continuation of this trial without end, where every trial is identical in probability structure and independent of all others.

A helpful way to visualize the idea is:

$$X_1, X_2, X_3, X_4, \dots$$

with each $X_i$ taking only values 0 or 1.

For example, if we toss a coin forever, the sequence might look like:

$$1, 0, 1, 1, 0, 0, 1, \dots$$

where each digit indicates success or failure.

Important features:

• The outcomes do not need to alternate
• Long runs of successes or failures may occur
• The probability rule remains the same at each step

The concept is extremely useful because it converts complex repeated experiments into a simple mathematical framework.

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2. Probability Structure of Infinite Bernoulli Trials

• In a Bernoulli sequence, the probability of any finite pattern can be calculated by multiplying probabilities because of independence.
• If $p$ is the probability of success and $q = 1-p$ is the probability of failure, then for a specific sequence of outcomes, the probability is the product of the probabilities of each outcome.

Example: If we want the probability of the first three trials being success, failure, success, then:

$$P(1,0,1) = p \cdot q \cdot p = p^2 q$$

If the trials are independent, this multiplication rule always holds.

For infinite sequences, we cannot assign probabilities to a complete endless list in the same simple way, but we can assign probabilities to events described by the first few trials or by patterns occurring somewhere in the sequence.

Examples of events in infinite Bernoulli trials:

• The first three trials are all successes
• At least one success occurs in the first five trials
• The sequence never contains a success
• Successes occur infinitely often

Important idea:

• Infinite sequences are studied through events
• Events can be finite or infinite in nature
• Independence makes calculations manageable

This structure is the basis for deeper results in probability, such as the law of large numbers and almost sure convergence.

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3. Long-Run Behavior and Events in Infinite Sequences

• A major reason infinite Bernoulli trials are studied is to understand what happens as the number of trials grows without bound.
• Even though each trial is random, patterns begin to appear in the long run.

One of the most important quantities is the relative frequency of success:

$$\frac{X_1 + X_2 + \cdots + X_n}{n}$$

This is the proportion of successes among the first $n$ trials.

If the trials are independent and identically distributed with success probability $p$, then the relative frequency tends to $p$ as $n$ becomes large, according to the law of large numbers.

Meaning:

• For a fair coin, the proportion of heads should get closer to $1/2$
• For a biased coin with $p=0.7$, the proportion of heads should get closer to $0.7$

Another important type of long-run event is:

Infinitely many successes

Only finitely many successes

Eventually all failures

Repeated occurrence of a pattern

Example: Consider tossing a fair coin forever.

• The probability of getting infinitely many heads is 1
• The probability of getting only finitely many heads is 0

This does not mean a head must appear at any particular moment, but rather that in the infinite process, heads continue to occur endlessly with probability 1.

This concept connects finite computations with infinite probabilistic behavior.

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Working / Process

1. Model each trial as a binary random variable

• Assign value 1 to success and 0 to failure.
• Ensure each trial has the same success probability $p$.

2. Check independence across trials

• The outcome of one trial must not affect any other trial.
• This allows probability calculations by multiplication.

3. Analyze finite events and extend to infinite behavior

• Compute probabilities of specific finite patterns.
• Study long-run properties such as relative frequency, repeated patterns, and infinite occurrence of events.

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Advantages / Applications

Simplifies repeated random experiments

• Many real-world situations can be reduced to a sequence of success/failure outcomes, making analysis easier.

Useful in statistics, reliability, and engineering

• Applied in coin tossing, defect testing, machine reliability, communication errors, and quality control.

Foundation for advanced probability theory

• Helps build understanding of laws of large numbers, convergence, stochastic processes, and random walks.

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Summary

• Infinite Bernoulli trials model endless repeated binary experiments with constant probability and independence.
• They are used to study both finite outcome patterns and long-run probabilistic behavior.
• Important terms to remember: Bernoulli trial, independence, success probability, failure probability, relative frequency