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MathidiaMathematics Academy
Statistical distributions
Lesson 53 of 75
Standard practice

The binomial model

Objective. Recognise and state the conditions for X~B(n,p).

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The binomial model describes the number of successes in a fixed number of independent trials, each with the same probability of success, written X~B(n,p). The four conditions needed are: a fixed number of trials n, each trial has only two outcomes, trials are independent, and the probability of success p is constant. For example, tossing a coin 10 times and counting heads satisfies B(10,0.5). If any condition fails, such as sampling without replacement from a small population, the binomial model should not be used.

Worked example

Which of these situations can be modelled by the binomial distribution X~B(n,p)?

Options
  • Tossing a fair coin 20 times and counting heads
  • Drawing 5 cards without replacement and counting aces
  • Selecting students until the first girl is found
  • Measuring heights of 20 random students
  1. 1Binomial needs fixed trials, two outcomes, constant p, independence
  2. 2Tossing a fair coin 20 times and counting heads satisfies all four
  3. 3Drawing cards without replacement changes p each time, so it fails
  4. 4Therefore the coin-tossing situation is binomial

Answer: Tossing a fair coin 20 times and counting heads

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Common mistake. Students apply the binomial model when probability changes between trials, such as drawing cards without replacement. Always check independence and constant p before stating X~B(n,p).

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