Playbook

Binomial Distribution

60 q - 1.27/paper - 22% HARD. A small, closed chapter with four subtopics and no surprises. Mean, Variance and Standard Deviation of a binomial variable (15 q) runs 13% HARD and is pure formula recall.

Questions in the bank
60
q/paper (2024–25 shifts)
1.27
Tagged HARD
22%
Subtopics
4

Strand: Quick-Win

When you’ll see it

A fixed number of independent trials with two outcomes and a constant success probability — coins, defective items, targets hit, questions guessed.

How this chapter is tested

60 q at 1.27 per paper and 22% HARD. A small, closed chapter with four subtopics and genuinely no surprises: the whole thing is one probability model and the four questions you can ask about it. That combination — low volume but low variance in what is asked — makes it one of the most reliable banks of marks on the paper.

Mean, Variance and Standard Deviation of a Binomial Variable is 15 q at 13% HARD and is pure formula recall: the mean is the number of trials times the success probability, and the variance multiplies that by the failure probability. The Binomial Setting and Probability Mass Function is 10 q at 10% HARD. That is 25 questions of near-free content in a 60-question chapter.

The two remaining blocks are Computing Binomial Probabilities (20 q, 30% HARD) and Parameter Estimation and the Probability Ratio (15 q, 27%). The second is the one students meet least often in school: you are given the mean and the variance, or a ratio of two consecutive probabilities, and asked to recover the number of trials and the success probability. It is short algebra once you have seen it, and worth an explicit drill.

Before any of that, check that the setting is actually binomial. Fixed number of trials, exactly two outcomes, independence between trials, and a success probability that does not change. Sampling without replacement fails the last two and the chapter's formulas do not apply.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • The binomial setting and probability mass function

    Verify the four conditions, then write the mass function with its combinatorial coefficient. 10 q at 10% HARD — the cheapest block and the gate for everything else.

  • Mean, variance, and standard deviation of a binomial variable

    Mean is trials times success probability; variance multiplies by the failure probability; standard deviation is its root. 15 q at 13% HARD and pure recall — bank it in one sitting.

  • Computing binomial probabilities

    Exactly-k, at-least-k and at-most-k probabilities, using the complement whenever the direct sum is longer. 20 q at 30% HARD, the largest block in the chapter.

  • Parameter estimation and the probability ratio

    Recover the number of trials and the success probability from the mean and variance, or from the ratio of two consecutive terms. 15 q at 27% HARD, and short algebra once the setup is familiar.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.

  • Variance taken as mean times success probability

    Variance uses the FAILURE probability as its third factor. The mean-times-p value is always among the options and is the most-taken wrong answer in this chapter's shape.

  • At-least-one computed as a direct sum

    At least one is one minus the probability of none, a single term. Attempting the full sum invites an arithmetic slip and burns far more than 1.8 minutes.

  • Combinatorial coefficient omitted

    The probability of a specific sequence is not the probability of that many successes. The uncounted value appears as a distractor and is smaller by exactly the coefficient.

  • Sampling without replacement treated as binomial

    Drawing without replacement changes the success probability between trials, so the model does not apply. The binomial answer is supplied anyway.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a per-subtopic mastery checkpoint. Read the notes once, then drill subtopic by subtopic below.

Binomial Distribution notes

Drill every binomial distribution question

60 questions from the bank, scoped to 4 bundled subtopics.

Drill the 60 questions

Drill one subtopic at a time

The 4 subtopics this playbook covers, in catalog order.

  • Computing Binomial ProbabilitiesDrill
  • Parameter Estimation and the Probability RatioDrill
  • Mean, Variance and Standard Deviation of a Binomial VariableDrill
  • The Binomial Setting and Probability Mass FunctionDrill

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