Section 6.2

The Binomial Probability Distribution

Analyze experiments with two possible outcomes—success or failure—and calculate the probability of specific results over multiple trials.

1

Identifying a Binomial Experiment

Not every probability experiment is binomial. To use the formulas in this section, an experiment must meet four specific criteria.

The 4 Conditions

  • 1
    Fixed Trials: The experiment is performed a fixed number of times ().
  • 2
    Independence: The trials are independent; the outcome of one does not affect the others.
  • 3
    Two Outcomes: Each trial has two mutually exclusive outcomes: Success or Failure.
  • 4
    Constant Probability: The probability of success () is the same for each trial.
Notation
Number of trials Probability of success Number of successes (0 to n)

Why "Success"?

"Success" just means the outcome we are tracking. Determining a "defective" part can be a "success" in statistical terms!

2

Binomial Probability Formula

Probability Distribution Function

LogicLens: Deconstructing the Formula

Part 1: The Counter ()

This counts the number of ways to arrange successes among trials. (e.g., getting 2 heads in 3 flips can happen as HHT, HTH, or THH).

Part 2: The Probability ()

This calculates the probability of one specific arrangement of successes and failures.

Binomial Probability Calculator
Condition
Probability
Exactly
0.00000
Fewer than
0.00000
At most
0.00000
More than
0.00000
At least
0.00000
3

Mean & Standard Deviation

For binomial distributions, calculating the mean and standard deviation is much simpler than the general discrete method.

Mean

The expected number of successes.

Standard Deviation

The spread/variation of successes.

Mean and Standard Deviation of a Binomial Random Variable
Mean ()
0.0000
Formula:
Std Dev ()
0.0000
Variance ()
0.0000
4

Distribution Shape

The shape of a binomial distribution depends on and . As increases, the distribution becomes more bell-shaped (Normal Approximation), especially when .

Binomial Distribution Visualizer
10
0.5
Properties
Mean ():5.00
Std Dev ():1.58
Shape:Symmetric
Adjust to see how the skewness changes. Notice that as increases, the shape becomes more symmetric (Normal Approximation).
Loading chart...
Skewed Right
Symmetric
Skewed Left

Common Pitfalls

Real-World Application

Quality Control

Manufacturers use the binomial distribution to accept or reject batches. If a batch has 5% defect rate, what is the probability that in a sample of 20, we find more than 2 defective items?

Medical Trials

If a new drug is 70% effective, determining the probability that it works for at least 8 out of 10 patients helps set expectation benchmarks.

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