Binomial Distribution Calculator (binompdf & binomcdf)

This free binomial distribution calculator works exactly like the TI-84 binompdf and binomcdf commands. Enter the number of trials, the probability of success and the number of successes, then pick exactly, at most, fewer than, at least, more than, or between. You get the probability, the full pdf and cdf table, a bar chart with your bars highlighted, the mean, variance and standard deviation, a normal approximation check, step-by-step working with nCr, and the exact TI-84 keystrokes for your own numbers. Results update as you type.

How to Use the Binomial Distribution Calculator

  1. Enter n, the number of trials. Any whole number up to 10,000,000 works.
  2. Enter p, the probability of success on one trial. Type a decimal (0.25), a fraction (1/4) or a percentage (25%).
  3. Choose the event. P(X = x), P(X ≤ x), P(X < x), P(X ≥ x), P(X > x) or P(a ≤ X ≤ b).
  4. Enter the number of successes x (and b for a range).
  5. Read the answer. The probability appears with the mean, variance and standard deviation. Open the steps, the TI-84 keys, the normal approximation or the full table below it.
  6. Copy or share. Copy results copies a summary with the TI-84 command. Copy share link copies a link that reopens the calculator with your numbers.

binompdf vs binomcdf: Which One Do You Need?

Most homework mistakes are off-by-one errors. Match the wording to the row below. The calculator builds the right command for you.

Question saysProbabilityTI-84 command
exactly xP(X = x)binompdf(n, p, x)
at most x, no more than xP(X ≤ x)binomcdf(n, p, x)
fewer than x, less than xP(X < x)binomcdf(n, p, x − 1)
at least x, no fewer than xP(X ≥ x)1 − binomcdf(n, p, x − 1)
more than x, greater than xP(X > x)1 − binomcdf(n, p, x)
between a and b (inclusive)P(a ≤ X ≤ b)binomcdf(n, p, b) − binomcdf(n, p, a − 1)

Binomial Distribution Formula

A binomial experiment has a fixed number of trials n, two outcomes per trial (success or failure), the same success probability p on every trial, and independent trials. Remember it as BINS: Binary, Independent, Number fixed, Same p. The number of successes X then follows the binomial distribution:

P(X = x) = nCr(n, x) · px · (1 − p)n − x

Here nCr(n, x) = n! / (x!(n − x)!) counts the ways to choose which x trials are the successes. px is the chance those trials all succeed, and (1 − p)n − x is the chance the rest all fail. Cumulative probabilities add these terms: P(X ≤ x) is the sum from 0 to x.

Worked Example: At Least 8 Successes

A basketball player makes 30% of her three-point shots. She takes 20 shots. What is the probability she makes at least 8?

  1. n = 20, p = 0.3, and we want P(X ≥ 8).
  2. “At least 8” is the complement of “7 or fewer”: P(X ≥ 8) = 1 − P(X ≤ 7).
  3. binomcdf(20, 0.3, 7) = 0.7723.
  4. P(X ≥ 8) = 1 − 0.7723 = 0.2277, about a 22.8% chance.
  5. Mean μ = 20 × 0.3 = 6 shots, σ = √(20 × 0.3 × 0.7) = 2.0494.

Press Example in the calculator to load this problem and see every step.

How to Do binompdf and binomcdf on a TI-84

  1. Press 2nd then VARS to open the DISTR menu.
  2. Choose A:binompdf( for exactly x, or B:binomcdf( for x or fewer. (On a TI-83 the letters are 0 and A.)
  3. On a TI-84 Plus CE or a newer TI-84 Plus with MathPrint, fill in the wizard: trials, p, x value. Select Paste and press ENTER twice.
  4. On older models the command pastes straight to the home screen. Type 20,0.3,7) and press ENTER.
  5. For “at least”, type 1 − before the command, as in 1−binomcdf(20,0.3,7).

Leave the x value blank and binompdf returns the whole list of probabilities, which is the same as the table in our calculator. Store it to L2 with STO→ 2nd 2 and put 0 to n in L1 to graph the distribution as a histogram. Practise these keys on our free TI-84 calculator online.

Normal Approximation to the Binomial

When n is large, the binomial looks like a bell curve. The usual check is np ≥ 10 and n(1 − p) ≥ 10. Then X is approximately normal with μ = np and σ = √(np(1 − p)). Because the binomial counts whole numbers, add a continuity correction of 0.5: P(X ≤ 7) becomes P(Y < 7.5), and P(X ≥ 8) becomes P(Y > 7.5).

The calculator runs this check for you, shows the corrected bounds, the matching normalcdf command and how far the approximation is from the exact answer. For more on bell curve areas, use our normal distribution calculator.

Why This Binomial Calculator Is Different

  • All six events in one place, each with the exact TI-84 command, including the 1 − binomcdf workarounds.
  • TI-84 keys with your own numbers, for both the MathPrint wizard and the classic home screen.
  • Full table and bar chart showing P(X = x), P(X ≤ x) and P(X ≥ x) for every x, with your bars highlighted.
  • Accurate for huge n. Probabilities are worked out in log space, so n = 1,000,000 or tiny tail areas do not round to 0.
  • Normal approximation check with continuity correction and the error versus the exact answer.
  • Step-by-step working with nCr, mean, variance and standard deviation.
  • Share links, dark mode and a phone-friendly layout.

Working with paired data instead? Try our linear regression calculator.

Binomial Distribution Calculator FAQ

binompdf is the binomial probability function on TI-83 and TI-84 calculators. binompdf(n, p, x) gives P(X = x), the probability of exactly x successes in n independent trials with success probability p.

binomcdf(n, p, x) gives the cumulative probability P(X ≤ x), the chance of x or fewer successes. It adds binompdf(n, p, 0) through binompdf(n, p, x).

P(X ≥ x) = 1 − binomcdf(n, p, x − 1). Subtract the cdf up to x − 1, not x, because x itself must stay in the answer. For “more than x”, use 1 − binomcdf(n, p, x).

binompdf gives one bar of the distribution (exactly x). binomcdf adds all the bars from 0 up to x (at most x). Use binompdf for “exactly” and binomcdf for “at most”, “no more than” and “fewer than”.

When np ≥ 10 and n(1 − p) ≥ 10 (some books use 5). Use μ = np and σ = √(np(1 − p)) and add or subtract 0.5 for the continuity correction. The calculator shows the check and both answers side by side.

Older models can overflow or run slowly for very large n. This calculator works in log space and handles n up to 10,000,000 without rounding to 0 or 1 too early.

Ti84CalcPro is not affiliated with or endorsed by Texas Instruments. TI-84 Plus CE is a trademark of Texas Instruments.