STAT1008 Chap.8 The Binomial Distribution
The Binomial Distribution
The Binomial Distribution models the number of successes in n independent trials with a constant success probability p. The chapter checks the four binomial properties against a story, applies the formula with combinations, uses the complement for “at least” questions, reads cumulative probabilities with BINOM.DIST, and summarises the distribution with mean np and variance np(1 − p).
It also covers derived success probabilities, combined categories and solving for the number of trials. Questions usually open by asking you to justify the binomial model, then request exact, at-least or between probabilities, the mean and standard deviation, and sometimes a judgement about whether an observed count is consistent with a stated success rate.
What this chapter covers
- 01
The four binomial properties
- 02
The binomial formula and combinations
- 03
At least and at most with the complement
- 04
Cumulative probabilities and BINOM.DIST
- 05
Mean and standard deviation of a binomial
- 06
Solving for the number of trials
Worked example · free
App users among ten customers
- 1X ~ Bin(10, 0.3): fixed n, two outcomes, constant p and independent customers.
- 2P(X = 3) = 10C3 (0.3)³(0.7)⁷ = 120 × 0.027 × 0.0823543 = 0.2668.
- 1P(X ≥ 2) = 1 − 0.0282 − 0.1211 = 0.8507.
- 1E(X) = np = 10 × 0.3 = 3 customers.
Key terms
- Binomial distribution
- The binomial distribution gives the number of successes in n independent trials with constant success probability p.
- Cumulative probability
- A cumulative probability is P(X ≤ x), the total probability of all values up to and including x.
- Complement rule
- The complement rule states that the probability of an event equals 1 minus the probability that it does not occur.
- Number of trials
- The number of trials n is the fixed count of independent observations in a binomial setting.
- Binomial trial
- A binomial trial is a single observation with two possible outcomes, success with probability p and failure with probability 1 minus p.
The Binomial Distribution FAQ
How do I show that a situation is binomial?
Check the four properties in the context of the question: a fixed number of trials, two outcomes on each trial, the same success probability every time, and independence between trials. Name each one with the scenario’s details.
How do I find at least 20 successes in Excel?
Use the complement with a cumulative call that stops one below the target: 1 minus BINOM.DIST(19, n, p, TRUE). Writing the inequality first, X at least 20 equals 1 minus P(X at most 19), avoids the off-by-one slip.
Why round up when solving for the number of trials?
A requirement such as a probability above 0.95 is met only by whole numbers of trials at or beyond the calculated bound. Rounding down would leave the probability just short of the target.
What is the difference between BINOM.DIST with TRUE and FALSE?
With FALSE, BINOM.DIST returns the probability of exactly that number of successes. With TRUE it returns the cumulative probability of that number or fewer, which is the quantity needed for at-most questions and, through the complement, for at-least ones.
What are the mean and variance of a binomial count?
For n trials with success probability p, the mean is np and the variance is np(1 minus p). Its standard deviation, the square root of np(1 minus p), is measured in the same units as the count, such as customers or defective items.
Exam move
Write the binomial formula from memory, then practise three question types: exact probabilities, complements for “at least” and ranges with two cumulative values. Check each in Excel with BINOM.DIST using FALSE and TRUE. Finish with one solve-for-n problem using logarithms, remembering that dividing by the log of a number below one reverses the inequality.
In the last week, mix binomial questions with normal ones so that choosing the right model becomes part of the practice. Make a short list of wording patterns and their inequalities, such as at least, more than, no more than and fewer than, and test yourself until the translation is instant. Then check one answer per session both by the formula and with BINOM.DIST to confirm the endpoint of each cumulative call.
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