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.

Final Thoughts

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.