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Exam Practice on Binomial Probability

Learn a repeatable approach for checking binomial conditions, choosing and showing calculations, and interpreting your results in context.

Intermediate 9 min read

What You'll Learn

  • Organize a binomial free-response response into State, Plan, Do, and Conclude steps.
  • Define the trial, success, \(n\), \(p\), and random variable \(X\) from a scenario.
  • Check each BINS condition using evidence from the problem.
  • Match exact, cumulative, and range events to appropriate binomial calculations.
  • Show a calculation and verify it with a second representation.
  • Write a conclusion that answers the question in context.

A Complete Binomial Response Does More Than Calculate

A binomial free-response problem asks you to connect a model to a context, not simply to enter values in a calculator. You need to define the count, decide whether a binomial model is reasonable, translate the requested event, calculate its probability, and explain what the result means.

Earlier tutorials, including Defining \(n\), \(p\), and \(X\) in Context, The BINS Checklist for Binomial Conditions, and Writing Binomial Probabilities in Proper Notation, established the tools for each of these decisions. Here, the new skill is coordinating them in a complete response. A useful structure is State, Plan, Do, Conclude: state the model and event, plan by checking conditions and choosing a calculation, do the calculation clearly, and conclude in context.

Free-response framework:
  • State: Define the trial, success, \(X\), \(n\), \(p\), and the event being asked about.
  • Plan: Check the binomial conditions and identify a calculation that matches the event.
  • Do: Show the formula or calculator command, the substitution, and the probability.
  • Conclude: State what the probability means for the situation.

Work from the Model to the Event

Start by translating the situation into a random variable. A trial is one repetition of the chance process. Define success as the outcome that adds one to the count \(X\); success does not have to be a favorable result. Then identify the fixed number of trials \(n\) and the probability \(p\) of success on each trial. The notation \(X\sim B(n,p)\) summarizes the proposed model.

Next, use BINS to justify that model. Check that each trial has two outcomes, the trials are independent, the number of trials is fixed, and the success probability is the same on each trial. Support each check with information from the scenario. When trials involve random sampling without replacement from a finite population, also check the 10% condition, as explained in The 10% Condition for Independence in Binomial Settings. A calculator can calculate a binomial probability, but it cannot establish that these conditions hold.

After checking the model, write the requested event using \(X\). “Exactly 3” is \(X=3\); “at most 3” is \(X\leq3\); and “at least 3” is \(X\geq3\). For a range, write both endpoints. Then select the calculation that matches: use \(\operatorname{binompdf}\) for an exact count, \(\operatorname{binomcdf}\) for a count at most an inclusive cutoff, and a complement or difference of cumulative probabilities for other events. These commands and their endpoints are covered in the earlier tutorials on binomial probabilities.

Worked Example: A Full Response with Two Requested Probabilities

Worked Example: A Full Response with Two Requested Probabilities

A lab uses a validation model in which each separately prepared test strip has probability 0.20 of giving a positive result. The outcomes of the strips are independent. A technician tests 10 strips. Let \(X\) be the number of strips that give positive results. Find the probability of exactly 3 positives and the probability of at least 2 positives. The lab reviews a batch if at least 2 strips are positive. What is the probability that this batch is reviewed?

State. One trial is testing one strip, and success is a positive result. The count \(X\) is the number of positive results among 10 strips. Thus \(n=10\), \(p=0.20\), and the proposed model is \(X\sim B(10,0.20)\). The events are \(X=3\) and \(X\geq2\); the review rule uses the second event.

Plan. Check BINS. Binary: each strip gives either a positive or a nonpositive result. Independent: the scenario states that the strip outcomes are independent. Number fixed: exactly 10 strips are tested. Same probability: the model assigns every strip the same positive-result probability, 0.20. These conditions support the binomial model. No finite-population sampling without replacement is described, so a 10% condition check is not needed. Use an exact binomial probability for \(X=3\); for \(X\geq2\), use the complement of \(X\leq1\).

Do: exactly 3 positives. The binomial formula gives:

$$ \begin{aligned} P(X=3) &=\binom{10}{3}(0.20)^3(0.80)^7\\ &=120(0.008)(0.2097152)\\ &=0.201326592\\ &\approx0.2013 \end{aligned} $$

As a calculator check, \(\operatorname{binompdf}(10,0.20,3)=0.201326592\), which agrees with the formula.

Do: at least 2 positives. The complement of \(X\geq2\) is \(X\leq1\), which includes zero or one positive result:

$$ \begin{aligned} P(X\geq2) &=1-P(X\leq1)\\ &=1-\operatorname{binomcdf}(10,0.20,1)\\ &=1-\left((0.80)^{10} +10(0.20)(0.80)^9\right)\\ &=1-(0.1073741824+0.268435456)\\ &=0.6241903616\\ &\approx0.6242 \end{aligned} $$

This result can also be checked by subtracting the two unwanted exact outcomes, zero and one positive, from 1. Their probabilities add to \(0.3758096384\), so \(1-0.3758096384=0.6241903616\), the same probability.

Conclude. Under the stated model, the probability of exactly 3 positive strips is about 0.2013. Because the lab reviews a batch when there are at least 2 positives, the probability that this batch is reviewed is about 0.6242.

Worked Example: Translate “At Most” Before Calculating

Worked Example: Translate “At Most” Before Calculating

A delivery service models each of 15 independently selected orders as having probability 0.10 of arriving late. Let \(X\) be the number of late orders. Find the probability that at most one of the 15 orders arrives late.

State. One trial is one order, success is an order arriving late, and \(X\) counts late orders. Therefore, \(n=15\), \(p=0.10\), and \(X\sim B(15,0.10)\). “At most one” means \(X\leq1\), including both zero and one late order.

Plan. Binary: an order arrives late or does not arrive late. Independent: the model states that the selected order outcomes are independent. Number fixed: 15 orders are considered. Same probability: each order has the same modeled late-arrival probability, 0.10. Thus BINS is supported by the scenario. There is no sampling without replacement from a finite population specified, so a 10% condition check is not relevant. Since the event is at most 1, use \(\operatorname{binomcdf}(15,0.10,1)\).

Do. The two included counts can also be added directly using the binomial formula:

$$ \begin{aligned} P(X\leq1) &=P(X=0)+P(X=1)\\ &=(0.90)^{15} +\binom{15}{1}(0.10)(0.90)^{14}\\ &=0.2058911321+0.3431518868\\ &=0.5490430189\\ &\approx0.5490 \end{aligned} $$

A calculator check is \(\operatorname{binomcdf}(15,0.10,1)=0.5490430189\). The cutoff is 1 because the event includes one late order; using a cutoff of 0 would omit that outcome.

Conclude. According to this model, the probability that at most one of the 15 orders arrives late is about 0.5490.

Worked Example: Show Both Endpoints for a Range

Worked Example: Show Both Endpoints for a Range

A seed-testing model assigns each seed a probability of 0.20 of germinating, independently of the other seeds. A fixed group of 9 seeds is tested. Let \(X\) be the number that germinate. Find the probability that between 1 and 3 seeds germinate, inclusive.

State. A trial is testing one seed, success is germination, and \(X\) counts germinated seeds. The model is \(X\sim B(9,0.20)\). The inclusive event is \(1\leq X\leq3\).

Plan. Binary: a seed germinates or does not germinate. Independent: independence is specified in the model. Number fixed: the group contains 9 seeds. Same probability: each seed has the same modeled probability, 0.20, of germinating. BINS is therefore supported. No finite-population sampling without replacement is described, so the 10% condition does not apply. For the inclusive range, subtract the cumulative probability through 0 from the cumulative probability through 3.

Do. Using the cumulative command and then checking by adding the exact probabilities gives:

$$ \begin{aligned} P(1\leq X\leq3) &=\operatorname{binomcdf}(9,0.20,3) -\operatorname{binomcdf}(9,0.20,0)\\ &=0.914358272-0.134217728\\ &=0.780140544\\ &\approx0.7801 \end{aligned} $$

For a direct check, the probabilities for exactly 1, 2, and 3 germinated seeds are \(0.301989888\), \(0.301989888\), and \(0.176160768\). Their sum is \(0.780140544\). Subtracting through 0 removes only the count below the requested range; subtracting through 1 would incorrectly remove the probability of exactly 1.

Conclude. Under the model, the probability that 1, 2, or 3 of the 9 seeds germinate is about 0.7801.

Common Mistakes and What a Strong Response Says

  • Giving a calculator command without establishing a model. A command returns a value even when its assumptions are unsupported. A strong response identifies evidence for each BINS condition before treating the result as a binomial probability.
  • Defining success without matching \(X\). If \(X\) counts late orders, late arrival is success, so \(p\) is the probability of a late order. State the definition so the choice of \(p\) is clear.
  • Using a cumulative cutoff that excludes a requested count. The final input to \(\operatorname{binomcdf}\) is inclusive. For “at most 4,” use 4; for “fewer than 4,” use 3.
  • Handling a range with the wrong lower cutoff. For \(a\leq X\leq b\), subtract the cumulative probability through \(a-1\), not through \(a\). That keeps the lower endpoint in the requested event.
  • Rounding too early or omitting the interpretation. Keep full calculator precision during intermediate steps, round the reported probability consistently, and name the event in the final sentence.
  • Concluding only with a number. “The probability is 0.6242” is incomplete if the reader cannot tell what that number describes. Say, for example, that under the model the probability the batch is reviewed is about 0.6242.

For full-credit communication, make the logic visible: define \(X\), state the model, support the conditions, express the event, show a matching calculation, and finish with an interpretation in context. A brief second check—such as adding exact probabilities or confirming a formula with a calculator command—can help catch a mismatched event or input.

AP Exam Tip: Treat each part of the prompt as a separate obligation. If asked to check conditions, name and address each one; if asked for a probability, show the event and matching calculation; if asked for a conclusion, answer the question in context rather than repeating the calculator output alone.

Key Takeaway

A binomial free-response solution is a connected argument: the conditions support the model, the definition of \(X\) determines \(n\) and \(p\), and the event determines the calculation. A contextual conclusion completes the response.

Key takeaway: Use State, Plan, Do, Conclude. Check BINS with evidence, translate the event before calculating, show enough work to make the probability verifiable, and interpret the result in the situation.

Check Your Understanding

For each item, outline the model, event, or communication needed for a complete response.

  1. A fixed group of 12 independent trials has probability 0.30 of success on each trial. Define \(X\), state its binomial model, and write the event “at least 4 successes.”
  2. For the setting in question 1, which calculator expression matches the event “at least 4 successes”? Explain why its cumulative cutoff is not 4.
  3. A response reports \(\operatorname{binompdf}(8,0.25,2)\) but never defines what \(X\) counts. What essential information is missing for a reader to interpret the result?
  4. For an inclusive range \(2\leq X\leq5\), which two cumulative probabilities should be subtracted, and why?
  5. In a free-response answer, what should the conclusion add beyond a rounded probability value?

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