This is a binomial probability: C(50,2) × (0.02)² × (0.98)⁴⁸. - Groen Casting
Understanding Binomial Probability: Decoding C(50,2) × (0.02)² × (0.98)⁴⁸
Understanding Binomial Probability: Decoding C(50,2) × (0.02)² × (0.98)⁴⁸
In the world of statistics, binomial probability models play a vital role in predicting outcomes in binary (success/failure) scenarios. One classic example is the expression C(50,2) × (0.02)² × (0.98)⁴⁸, which represents a specific binomial probability scenario. In this article, we break down what this formula means, how it applies to real-world problems, and why it’s essential for both students, data analysts, and decision-makers.
Understanding the Context
What Is a Binomial Probability?
Binomial probability models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success (with probability p) or failure (with probability 1 − p). The formula for binomial probability is:
> P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ
Where:
- n = number of trials
- k = number of successes
- p = probability of success per trial
- C(n, k) is the combination of n items taken k at a time
Key Insights
Breaking Down the Expression: C(50,2) × (0.02)² × (0.98)⁴⁸
This expression represents a binomial probability calculation with:
- n = 50 — total number of independent trials (e.g., tests, observations)
- k = 2 — number of desired successes
- p = 0.02 — probability of success on a single trial (e.g., passing a test, defect in manufacturing)
- C(50,2) — number of ways to achieve 2 successes in 50 trials
Calculating each component:
1. C(50, 2) — Combinations:
C(50,2) = 50! / [2!(50−2)!] = (50 × 49) / (2 × 1) = 1225
This means there are 1,225 different ways to select 2 successful outcomes among 50 trials.
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2. (0.02)² — Probability of 2 successes:
(0.02)² = 0.0004
This represents the probability that exactly 2 trials result in success, assuming independence.
3. (0.98)⁴⁸ — Probability of 48 failures:
(0.98)⁴⁸ ≈ 0.364 (calculated using exponentiation)
This reflects the likelihood that the remaining 48 trials result in failure.
Full Probability Value
Putting it all together:
P(X = 2) = 1225 × 0.0004 × 0.364 ≈ 1225 × 0.0001456 ≈ 0.1786
So, there’s approximately a 17.86% chance of getting exactly 2 successes in 50 trials with a 2% success rate per trial.
Real-World Applications
1. Quality Control in Manufacturing
If a product defect occurs in 2% of items and a batch of 50 items is randomly inspected, this model estimates the probability of finding exactly 2 defective items.
2. Medical Testing & Epidemiology
In rare disease screening or survey sampling, understanding the chance of observing a very low number of positive outcomes under low prevalence is crucial.
3. Quality Assurance in Software Testing
When testing software, a 2% failure rate per test cycle, over 50 cycles, lets QA teams estimate the likelihood of catching exactly 2 bugs — informing resource allocation and risk assessment.
