Probability is the mathematical language of uncertainty. When events happen more than once—especially under similar conditions—their analysis falls under repeated probability. Whether you’re rolling dice, flipping coins, testing software, or making long-term business decisions, understanding repeated probability is essential.
Repeated Probability Calculator
The calculator assumes independent trials.
In this post, we’ll explain everything you need to know about repeated probability: its principles, calculations, real-world applications, common mistakes, and how it differs from other probability types.
1. What Is Repeated Probability?
Repeated probability refers to the likelihood of one or more outcomes occurring across multiple independent or dependent trials of the same event.
In simple terms, it’s answering questions like:
- What’s the chance of flipping heads three times in a row?
- What’s the probability of passing five exams consecutively?
- What’s the chance of a machine failing twice in a month?
Two Key Types:
| Type | Description | Example |
|---|---|---|
| Independent Trials | Each event does not affect the next | Coin tosses, dice rolls |
| Dependent Trials | Each event may affect the next | Drawing cards without replacement |
2. The Mathematics of Repeated Probability
2.1 Independent Events
If you repeat an event n times, and each event has probability p, then:
- All Successes:
P(success in all n trials) = pⁿ - All Failures:
P(failure in all n trials) = (1 – p)ⁿ
Example:
What is the probability of flipping a fair coin and getting heads 3 times in a row?
P = (0.5) × (0.5) × (0.5) = 0.125
2.2 Mixed Outcomes
You can also calculate the probability of getting a certain number of successes in n repeated trials using the binomial distribution: P(k successes in n trials)=C(n,k)⋅pk⋅(1−p)n−kP(k \text{ successes in } n \text{ trials}) = C(n, k) \cdot p^k \cdot (1 – p)^{n – k}P(k successes in n trials)=C(n,k)⋅pk⋅(1−p)n−k
Where:
- C(n, k) is the number of combinations (n choose k)
- p is the probability of success
- k is the number of desired successes
Example:
What’s the chance of getting exactly 2 heads in 3 flips? P=C(3,2)×(0.5)2×(0.5)1=3×0.25×0.5=0.375P = C(3, 2) × (0.5)^2 × (0.5)^1 = 3 × 0.25 × 0.5 = 0.375P=C(3,2)×(0.5)2×(0.5)1=3×0.25×0.5=0.375
3. Dependent Repeated Events
When events are not independent (e.g., drawing cards without replacement), each outcome affects the next.
Example:
A bag has 3 red and 2 blue balls. What’s the probability of drawing 2 red balls in a row without replacement?
- P(first red) = 3/5
- P(second red | first red) = 2/4 = 1/2
- Total: P = 3/5 × 1/2 = 0.3
4. Applications of Repeated Probability
Repeated probability is used everywhere:
4.1 Games and Gambling
| Situation | Use |
|---|---|
| Dice games | Repeated rolls and score predictions |
| Poker | Repeated draws, bluff odds |
| Lottery | Multiple number draws |
4.2 Quality Control
Manufacturers estimate the chance of producing defective items over time or products that fail after repeated use.
4.3 Medicine
Researchers use repeated probability in:
- Clinical trials
- Vaccine effectiveness across multiple doses
- Probability of side effects in repeated treatments
4.4 Sports
Analysts calculate win streaks, batting averages, or the chance a team wins 3 games in a row using repeated trials.
5. Tables for Common Repeated Probabilities
Table 1: Probability of Repeated Coin Tosses (Fair Coin)
| Tosses | All Heads | All Tails | 1 Head & 1 Tail |
|---|---|---|---|
| 1 | 0.5 | 0.5 | N/A |
| 2 | 0.25 | 0.25 | 0.5 |
| 3 | 0.125 | 0.125 | 0.75 |
Table 2: Repeated Success in Trials (p = 0.8)
| Trials | 1 Success | 2 in a Row | 3 in a Row |
|---|---|---|---|
| Probability | 0.8 | 0.8² = 0.64 | 0.8³ = 0.512 |
6. Real-World Examples
6.1 Software Testing
Imagine a program has a 5% chance of crashing per test run. What’s the chance it crashes at least once in 10 repeated runs?
- P(no crash) = (0.95)¹⁰ ≈ 0.5987
- So, P(at least one crash) = 1 – 0.5987 = 0.4013
6.2 Medical Screening
A test has a 90% accuracy rate. If you test a patient 3 times, what’s the chance all results are correct?
P = 0.9³ = 0.729
7. Common Mistakes in Repeated Probability
| Mistake | Why It Happens | Example |
|---|---|---|
| Assuming independence when it’s not | Events affect each other | Drawing cards without replacing |
| Forgetting to square or cube the probability | Misapplying basic rules | Thinking P(A and A) = P(A) |
| Ignoring “at least once” phrasing | Overlooking complement rule | Miscalculating at least one success |
| Rounding too early | Leads to inaccurate answers | Truncating values mid-calculation |
8. Special Repeated Scenarios
8.1 At Least One Success
For repeated independent trials: P(at least one success)=1−(1−p)nP(\text{at least one success}) = 1 – (1 – p)^nP(at least one success)=1−(1−p)n
Example:
What’s the probability of getting at least one 6 in 4 rolls of a fair die?
- p = 1/6
- P(no 6) = (5/6)⁴ ≈ 0.4823
- P(at least one 6) = 1 – 0.4823 = 0.5177
8.2 First Success on the k-th Trial (Geometric Distribution)
The probability of having your first success on the k-th trial: P(k)=(1−p)k−1⋅pP(k) = (1 – p)^{k – 1} \cdot pP(k)=(1−p)k−1⋅p
Example:
What is the chance of getting the first heads on the 3rd toss?
- p = 0.5
- P = (0.5)² × 0.5 = 0.125
9. Graph: Repeated Success vs. Number of Trials
A visual of how repeated successes become less likely with more trials if probability < 1:
| Number of Trials (n) | P(successⁿ) if p = 0.9 |
|---|---|
| 1 | 0.9 |
| 2 | 0.81 |
| 5 | 0.59049 |
| 10 | 0.34867 |
As the number of repetitions increases, overall success probability drops unless p is 1.
10. Repeated Probability in Simulation
In real-world analysis, we often simulate repeated trials using:
- Monte Carlo simulations
- Python/R programming
- Spreadsheet models
These are used in industries like:
- Insurance risk modeling
- Stock market forecasting
- Engineering reliability analysis
11. Repeated Events in Dependent Systems
Not all repeated probabilities are simple. Consider:
- Markov chains: Where future probabilities depend on the current state.
- Bayesian updates: Repeated evidence changes the probability distribution.
- Machine learning models: Repeated training data adjusts confidence over time.
12. Summary Table: Key Formulas
| Scenario | Formula |
|---|---|
| All successes (independent) | P = pⁿ |
| All failures (independent) | P = (1 – p)ⁿ |
| At least one success | P = 1 – (1 – p)ⁿ |
| Exactly k successes in n trials | Binomial: C(n,k) * pᵏ * (1-p)ⁿ⁻ᵏ |
| First success on k-th trial | Geometric: (1 – p)ᵏ⁻¹ * p |
13. Final Thoughts
Repeated probability is one of the most practical and widely applicable topics in all of probability theory. It helps you answer questions like:
- How reliable is something over time?
- What’s the chance of repeating success?
- How likely are rare events over many tries?
Whether you’re analyzing product performance, clinical trials, or simply planning your odds in a game, mastering repeated probability will give you a powerful decision-making edge.