Check Four Conditions First. The Formula Is the Easy Part

You'll learnthe four conditions a setting must pass before the binomial model applies, why the combination coefficient is part of the formula, the mean np and standard deviation √(np(1−p)), and how to translate at least, more than, and at most into the right sum.

A player makes 70% of her free throws and takes five shots. Five trials, two outcomes each, the same chance every time, no shot affecting another — the setting passes all four binomial conditions, so the count of makes has a distribution the formula can write down. This animation builds that distribution twice: once from the formula, and once by listing all 32 ways five shots can go and weighing each one. The two agree at every count, which is what makes the combination coefficient visible rather than memorized — ten of the 32 sequences produce exactly three makes, and dropping the ten is a factor-of-ten error. Then the boundary words: at least 4 collects two bars, more than 4 keeps one, and the complement of at least 3 is at most 2, not at most 3. It ends where most real mistakes start: a club of fifteen where the model never applied at all.

8 STEPS · 6 QUICK CHECKS · THE BINOMIAL SETTING · np AND √(np(1−p)) · BOUNDARY WORDS · v1

n = 5 shots at p = 0.7 · 32 sequences counted · four conditions first

Before you start

What you're looking at

Five free throws at a 70% success rate — first as all 32 possible make-miss sequences, then as the probability distribution of the number of makes.

The question

When does the binomial model apply, what does the combination coefficient actually count, and which bars does a boundary word like "at least 4" collect?

Watch for

The ten outlined sequences with exactly three makes — the 10 in the formula — and the final step, where a club of fifteen fails the conditions and the model's answer is measurably wrong.

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