Binomial Probability
P(X = k) = C(n,k) ร pแต ร (1 - p)โฟโปแต
The binomial probability formula calculates the probability of getting exactly k successes in n independent Bernoulli trials, each with the same probability of success p. It is one of the most important discrete probability distributions in statistics.
Variables
The probability of exactly k successes
The fixed number of independent trials
The desired number of successes (0 โค k โค n)
The probability of success on each individual trial
Example Calculation
Scenario
A fair coin is flipped 10 times. What is the probability of getting exactly 7 heads?
Given Data
Calculation
P(X = 7) = C(10,7) ร (0.5)โท ร (0.5)ยณ = 120 ร 0.0078125 ร 0.125
Result
P(X = 7) = 0.1172
Interpretation
There is approximately an 11.72% chance of getting exactly 7 heads in 10 flips of a fair coin. While 7 heads is more than expected (5), it is not extremely unlikely.
When to Use This Formula
- โCalculating probabilities for a fixed number of independent yes/no trials
- โQuality control problems where items are classified as defective or not defective
- โSurvey analysis where each respondent has the same probability of a particular response
- โModeling the number of successes in repeated Bernoulli experiments
Common Mistakes
- โForgetting to include the binomial coefficient C(n,k) which counts the number of arrangements
- โUsing the binomial distribution when trials are not independent
- โConfusing P(X = k) with cumulative probabilities P(X โค k)
- โApplying the binomial model when the probability of success changes between trials
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Common questions about this formula
There must be a fixed number of trials (n), each trial must be independent, each trial must have exactly two outcomes (success or failure), and the probability of success (p) must be the same for every trial. These are sometimes called the BINS conditions.
To find P(X โค k), sum the individual probabilities P(X = 0) + P(X = 1) + ... + P(X = k). For large n, you can approximate the binomial distribution with the normal distribution using ฮผ = np and ฯ = โ(np(1-p)), applying a continuity correction.