The TI-84 has two built-in functions for binomial distribution problems: binompdf for finding the probability of exactly one outcome, and binomcdf for finding the probability of a range of outcomes. Both are in the same menu, and once you know which one to use, the actual button presses take about ten seconds.
Finding the Right Menu
Both binomial functions live in the distributions menu. Press 2nd, then VARS (this key is labeled “DISTR” above it). Scroll down until you see binompdf and binomcdf. On newer TI-84 Plus CE models, you can also press ALPHA then APPS after opening the DISTR menu to jump directly to binomcdf (option B) without scrolling.
If your calculator has Stat Wizards turned on (the default on newer models), selecting either function will pop up a fill-in-the-blank screen that labels each input for you. If you prefer the classic command-line style where you type everything on one line, go to MODE, scroll to “Stat Wizards,” and switch it to OFF.
What Each Function Does
Both functions need the same three inputs, entered in the same order: the number of trials (n), the probability of success on a single trial (p), and the number of successes you care about (x).
- binompdf(n, p, x) calculates the probability of getting exactly x successes. Use this when a problem says “exactly,” such as “What is the probability of getting exactly 4 heads in 7 flips?”
- binomcdf(n, p, x) calculates the probability of getting x or fewer successes. It adds up every probability from 0 through x. Use this when a problem says “at most,” “no more than,” or “fewer than” (with a small adjustment explained below).
Calculating an Exact Probability
Suppose you flip a coin 7 times and want the probability of getting exactly 4 heads, where the probability of heads on each flip is 0.35.
Press 2nd, then VARS to open the DISTR menu. Select binompdf. Enter the values separated by commas: 7, 0.35, 4. Close the parenthesis and press ENTER. Your screen should read binompdf(7, .35, 4) and the calculator returns the probability.
Calculating “At Most” Probabilities
If the question asks for the probability of 4 or fewer successes out of 7 trials with p = 0.35, use binomcdf instead. The steps are identical: press 2nd, VARS, select binomcdf, and enter binomcdf(7, .35, 4). The calculator adds up P(0) + P(1) + P(2) + P(3) + P(4) and gives you the cumulative result.
For “fewer than” questions, adjust x down by one. “Fewer than 5 successes” means 4 or fewer, so you’d enter binomcdf(7, .35, 4).
Calculating “At Least” Probabilities
“At least” problems require a small workaround because binomcdf only counts from the bottom up. The trick is to use the complement: P(at least x) = 1 minus P(at most x − 1).
For example, say you have 6 trials with p = 0.2 and want the probability of at least 2 successes. That equals 1 minus the probability of 0 or 1 success. On the calculator, type 1 − binomcdf(6, .2, 1) and press ENTER. The result is approximately 0.34464.
The button sequence: press 1, then the minus key, then 2nd, VARS, select binomcdf, enter 6, .2, 1, close the parenthesis, and press ENTER.
Calculating a Range of Values
Some problems ask for the probability of getting between two values, like “between 2 and 4 successes, inclusive.” The approach is to subtract one cumulative probability from another: binomcdf up to the top of the range minus binomcdf up to one below the bottom of the range.
Using the same example of 6 trials with p = 0.2, the probability of 2 to 4 successes is binomcdf(6, .2, 4) − binomcdf(6, .2, 1). You can type this as one expression. Press 2nd, VARS, select binomcdf, enter 6, .2, 4, close the parenthesis, press minus, then repeat 2nd, VARS, binomcdf, enter 6, .2, 1, close the parenthesis, and press ENTER. The result is approximately 0.34304.
Quick Reference for Wording
- “Exactly x” → binompdf(n, p, x)
- “At most x” or “x or fewer” → binomcdf(n, p, x)
- “Fewer than x” → binomcdf(n, p, x−1)
- “At least x” or “x or more” → 1 − binomcdf(n, p, x−1)
- “More than x” → 1 − binomcdf(n, p, x)
- “Between a and b, inclusive” → binomcdf(n, p, b) − binomcdf(n, p, a−1)
Fixing Common Errors
If the calculator returns ERR:DOMAIN, one of your inputs is outside the allowed range. The number of trials must be a positive whole number, the probability p must be between 0 and 1, and the x value must be a whole number between 0 and n. Double-check that you didn’t accidentally enter a probability greater than 1 or a negative number of successes.
If you see ERR:ARGUMENT, the calculator received the wrong number of inputs. This usually means a missing comma or an extra parenthesis. Make sure you have exactly three values separated by two commas, and that the expression ends with a closing parenthesis before you press ENTER.
One other easy mistake: entering a decimal for the number of trials. If your problem says 10 trials, enter 10, not 10.0. The calculator expects an integer for n and may throw a domain error otherwise.