Binomial Distribution on TI-84 and Desmos
Binomial probability questions ask for “exactly”, “at most”, “at least” or “between”. Each one maps to a specific calculator command, and the “at least” case is where most mistakes happen. This guide covers the TI-84 and the Desmos calculator in Bluebook, with one worked example you can follow on either.
binompdf is for exactly k. binomcdf adds up everything from 0 to k, so it is at most k. For at least k, use 1 minus the cdf at k − 1.First: is it really binomial? (BINS)
- Binary: each trial is a success or a failure.
- Independent: one trial does not affect the next. Sampling without replacement is fine if the sample is at most 10% of the population.
- Number of trials is fixed in advance.
- Same probability of success on every trial.
The commands
| You need | TI-84 (DISTR = 2nd VARS) | Desmos |
|---|---|---|
| Exactly k | binompdf(n, p, k) | binomialdist(n,p).pdf(k) |
| At most k (≤) | binomcdf(n, p, k) | binomialdist(n,p).cdf(k) |
| Fewer than k (<) | binomcdf(n, p, k−1) | binomialdist(n,p).cdf(k-1) |
| At least k (≥) | 1 − binomcdf(n, p, k−1) | 1-binomialdist(n,p).cdf(k-1) |
| More than k (>) | 1 − binomcdf(n, p, k) | 1-binomialdist(n,p).cdf(k) |
| Between a and b, inclusive | binomcdf(n, p, b) − binomcdf(n, p, a−1) | binomialdist(n,p).cdf(a,b) |
In Desmos, just type binomialdist and it formats itself. The TI-84 commands are in the DISTR menu (2nd then VARS).
Worked example: free throws
A player makes 60% of free throws and takes 20 shots. Let X be the number made, so X is binomial with n = 20 and p = 0.6.
| Question | Calculator input | Answer |
|---|---|---|
| Exactly 12 made | binomialdist(20,0.6).pdf(12) or binompdf(20, 0.6, 12) | 0.1797 |
| At most 9 made | binomialdist(20,0.6).cdf(9) or binomcdf(20, 0.6, 9) | 0.1275 |
| At least 15 made | 1-binomialdist(20,0.6).cdf(14) or 1 − binomcdf(20, 0.6, 14) | 0.1256 |
| Between 10 and 14 made | binomialdist(20,0.6).cdf(10,14) | 0.7469 |
The expected number made is μ = np = 20(0.6) = 12, and the standard deviation is σ = √(np(1−p)) = √(20 × 0.6 × 0.4) = 2.19.
Check the exactly-12 answer by hand with P(X = k) = C(n, k) pk(1 − p)n−k: C(20, 12) = 125,970, so P(X = 12) = 125,970 × 0.612 × 0.48 = 0.1797. The combinations calculator will find the C(20, 12) part for you.
Translate the words into a symbol
- “At most”, “no more than”, “k or fewer” mean ≤ k.
- “Fewer than”, “less than” mean < k, which is ≤ k − 1.
- “At least”, “no fewer than”, “k or more” mean ≥ k.
- “More than”, “greater than”, “exceeds” mean > k, which is ≥ k + 1.
Whenever the question says “at least” or “more than”, you are using the complement: everything except the values you don’t want. That is why the cdf is evaluated one step below k for “at least”.
What to write on the free-response section
Define the variable, name the distribution with its parameters, and show the calculation with labels. For the “at least 15” question:
“Let X = the number of free throws made in 20 attempts. X is binomial with n = 20 and p = 0.6. P(X ≥ 15) = 1 − P(X ≤ 14) = 1 − binomcdf(20, 0.6, 14) = 0.1256.”
The 2026 scoring guidelines accepted labeled Desmos syntax in the same way, such as binomialdist(n = 10, p = 0.2459) for a different problem, alongside TI notation. A bare number with no setup is not enough.
Common mistakes
- Using
binomcdfwhen the question says “exactly”. The cdf includes every value up to k. - Off by one on “at least”. P(X ≥ 15) is 1 − P(X ≤ 14), not 1 − P(X ≤ 15).
- Skipping the BINS check. If the trials are not independent or p changes, the binomial model does not apply.
- Mixing up p and 1 − p. p is the probability of the outcome you are counting as a success.
Everything else in this unit is in the Unit 2 study guide. The same calculators and the rest of the procedures are in the AP Stats calculator guide and the one-page cheat sheet.
Desmos syntax tested in the College Board practice calculator and checked against independent statistical software. Updated for the May 2027 exam. Last reviewed October 2026.
Frequently Asked Questions
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How do I calculate a binomial probability on a TI-84?
Press 2nd then VARS to open the DISTR menu. Choose binompdf for the probability of exactly k successes, entering n, p and k, or choose binomcdf for the probability of k or fewer successes, entering n, p and k.
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How do I find "at least" in a binomial problem on Desmos?
Use the complement. For at least k successes, type 1-binomialdist(n,p).cdf(k-1). For example, the probability of at least 15 successes in 20 trials with p = 0.6 is 1-binomialdist(20,0.6).cdf(14), which is about 0.1256.
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What is the difference between binompdf and binomcdf?
The pdf gives the probability of exactly k successes. The cdf is cumulative: it gives the probability of k or fewer successes by adding the probabilities for 0 through k. In Desmos these are the .pdf and .cdf properties of binomialdist.
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When can I use the binomial distribution?
When each trial has two outcomes, the trials are independent, the number of trials is fixed, and the probability of success is the same each time. When sampling without replacement, the sample should be no more than 10% of the population so independence is a reasonable assumption.