Step by Step
B
Binary outcomes
Each trial must have only two possible outcomes — typically labeled "success" and "failure."
I
Independent trials
The outcome of one trial must not affect the outcome of any other trial.
N
Fixed Number of trials (n)
The total number of trials must be fixed and known in advance.
S
Same probability (p) each trial
The probability of success must stay the same across every trial.
Applied Walkthrough
1
A basketball player takes 10 free throws, each with a 70% chance of success, and each shot is independent of the others.
2
Check the BINS conditions: Binary (make or miss), Independent (each shot doesn't affect the next), fixed Number (n=10), Same probability (p=0.7 each time). All four conditions hold, so this is a binomial situation.
3
Calculate the mean: μ = np = 10(0.7) = 7 — the expected number of made shots out of 10.
4
Calculate the standard deviation: σ = √(np(1−p)) = √(10×0.7×0.3) = √2.1 ≈ 1.45 — describing the typical variability around that expected value of 7 made shots.
Exam Application
Exams test whether you can verify all four BINS conditions before applying binomial formulas, and whether you can correctly calculate the mean (np) and standard deviation (√(np(1−p))) of a binomial distribution.
⚠ Common Trap
The most common trap is applying binomial formulas without first verifying all four BINS conditions hold — particularly the independence condition, which is sometimes violated (for example, sampling without replacement from a small population).
✓ Quick Self-Check
1. What does the BINS mnemonic stand for?
Binary outcomes, Independent trials, fixed Number n, Same probability p each trial.
Tap to reveal / hide
2. What is the formula for the mean of a binomial distribution?
μ = np.
Tap to reveal / hide
3. What is the formula for the standard deviation of a binomial distribution?
σ = √(np(1−p)).
Tap to reveal / hide
4. For 10 free throws with a 70% success rate each, what is the expected (mean) number of makes?
7.
Tap to reveal / hide
5. When can the normal distribution be used to approximate a binomial distribution?
When both np and n(1−p) are at least 10.
Tap to reveal / hide