Using binomcdf for At Most Probabilities

From Exact Probabilities to Cumulative Probabilities

In Using binompdf for Exact Probabilities, you learned to use binompdf to find the probability that a binomial random variable equals one particular value. But many questions ask about several possible counts at once. For example, a packaging team might want the probability that at most two items in a batch are damaged. That event includes zero, one, or two damaged items—not just exactly two.

The command binomcdf finds this kind of cumulative probability for a binomial random variable. The letters “cdf” stand for cumulative distribution function. In this setting, the command adds the probabilities from zero successes through a specified upper cutoff. This is the calculator version of the cumulative probabilities for discrete variables discussed earlier in the course.

Each term in the sum is an exact probability that could be calculated with the binomial formula or binompdf. binomcdf does the addition for you. The input order is \(n\), then \(p\), then \(k\), just as binompdf takes \(n\), \(p\), and an exact count. On a TI-84-style calculator, the command is generally available through the distribution menu; menu paths can vary by model.

Translate “At Most” Before You Calculate

As in Defining n, p, and X in Context, first identify what one trial is, what counts as success, and what \(X\) counts. Then translate the wording into an inequality. “At most \(k\)” means \(X\leq k\): the cutoff is included, and all smaller possible values are included too. For a count of successes, those values start at zero.

Do not confuse the exact and cumulative commands. \(\operatorname{binompdf}(n,p,k)\) gives \(P(X=k)\), while \(\operatorname{binomcdf}(n,p,k)\) gives \(P(X\leq k)\). The same cutoff value appears in each command, but the events are different. In particular, “at most two” includes \(X=0\), \(X=1\), and \(X=2\).

As with binompdf, binomcdf evaluates a model; it does not verify that the model is appropriate. Use the BINS checklist from the earlier tutorials. The trials need binary outcomes, independence, a fixed number of trials, and the same probability of success on every trial. If any of these conditions is not reasonable, the binomial calculation may not describe the situation well.

Worked Example: At Most Two Damaged Packages

The exact-probability check confirms the cumulative meaning: the result is a sum of three probabilities. Using only binompdf for \(k=2\) would give \(P(X=2)=0.2458\), which answers “exactly two,” not “at most two.”

Worked Example: At Most Four Successful Checks

The upper cutoff is included: the term for \(X=4\) is part of the cumulative probability. Leaving out that term would instead calculate \(P(X\leq 3)\). Always connect the calculator’s last input to the endpoint in the inequality.

Worked Example: At Most One Response

This example has only two terms in its cumulative sum, but the same command works when many counts are included. The calculator adds all the exact probabilities up to the cutoff; you do not need to enter each count separately.

Common Mistakes and AP Exam Tip

A full-credit response makes the event and the calculator command agree. Show which random variable is being counted, translate the wording into an inequality, and state what the result means in the situation.

• Using binompdf for an at-most event. binompdf gives the probability of one exact count. For “at most four,” use binomcdf with an upper cutoff of 4.
• Leaving out smaller counts. \(P(X\leq k)\) includes every possible value from 0 through \(k\), not just \(k\) and not just values close to \(k\).
• Excluding the endpoint. “At most four” includes four. Write \(X\leq 4\) before entering the command to keep the endpoint clear.
• Reversing the inputs. Enter \(n\), \(p\), and then \(k\). Do not enter the number of failures or the failure probability unless the random variable and success have been defined differently.
• Treating a calculator result as proof of a valid model. Check BINS using the context. A correct command cannot repair a model with dependent trials or changing success probabilities.
• Giving only a decimal. State the event and interpret the answer. For example, say “The model gives a probability of about 0.5767 that zero or one reminder receives a response,” rather than writing only “0.5767.”

As a quick reasonableness check, a cumulative probability cannot be smaller than any one exact probability it includes. Also, increasing the upper cutoff can only keep the cumulative probability the same or make it larger, because the event includes at least as many possible counts.

Key Takeaway

binomcdf finds a cumulative binomial probability. Its final input is an inclusive upper cutoff: the result adds the probabilities for zero successes, one success, and every possible count up through that cutoff.