Binomial Distribution

The binomial distribution is an example of a discrete random variable. It has two parameters n (number of trials) and p (probability of success of one trial): X~B(n , p). For a situation to be described using a binomial model, the following must be true


  • A finite number of trials, n are carried out
  • The trials are independent
  • The outcome of each trial is deemed either a success or a failure
  • The probability, p, of a successful outcome is the same for each trial

Key Concepts

On this page, you should learn about

  • the binomial distribution
  • mean of the binomial distribution
  • variance of the binomial distribution
  • the formula for calculating probabilities

P(X = r) = nCr x pr x (1 - p)n - r

Summary

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Exam-style Questions

Question 1

On average, it is found that 5% of AirPods* made on a production line are faulty.

a) Find the probability that in a random sample of 10, there are

i) No faulty AirPods

ii) more than one faulty set of AirPods

b) A sample of n sets of AirPods is taken from the production line. If the probability that there is at least one faulty AirPod is more than 75%, find the smallest possible value of n

* Fictitious data. I’m sure that AirPods production is very reliable!


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Question 2

Monster Energy drink cans are advertised to contain 500ml. During the production, the volume X in ml in cans of the drink can be modelled by a normal distribution with mean 504 and variance 10.

a) For a randomly selected can, work out P(X > 500).

b) Cans are put in packs of 6. Find the probability that at least 5 cans are have a volume of at least 500ml.


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Question 3

A glass contains 5 green sweets and m sweets of other colours. A sweet is taken at random.

a) Write down the probability that the sweet is green.

b) The sweet is replaced in the glass and the process is repeated a further three times. Each time, it is noted whether a green sweet is taken. The variance of the number of green sweets taken over the whole process in calculated to be 0.75.

Work out how many sweets in total there are in the glass.


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