The binomial distributionEdexcel International A Level Further Maths: Subtopic test

1. 1
   The discrete random variable $X$ has the binomial distribution $\mathrm{B}(12,\,0.25)$.

   (a)

   Find $\mathrm{P}(X=3)$.

   [1 mark]

   • A$0.00117$
   • B$0.649$
   • C$0.258$
   • D$0.00035$

   (b)

   Find $\mathrm{Var}(X)$.

   [1 mark]

   • A$2.25$
   • B$3$
   • C$1.5$
   • D$9$

   (c)

   Find $\mathrm{P}(X\ge2)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A machine makes components. Each component is independently defective with probability $0.04$. A random sample of $20$ components is taken, and the number of defective components in the sample is $X$.

   (a)

   Find the probability that exactly one component in the sample is defective.

   [1 mark]

   • A$0.0184$
   • B$0.810$
   • C$0.800$
   • D$0.368$

   (b)

   Which statement is an assumption needed for $X$ to be modelled by a binomial distribution?

   [1 mark]

   • AThe mean number of defectives equals the variance
   • BWhether one component is defective does not affect whether any other is
   • CThe probability of a defect is different for each component
   • DThe number of components in the sample is random

   (c)

   Find the probability that more than $2$ components in the sample are defective.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A multiple-choice test has $15$ questions. Each question has four options, of which exactly one is correct. A student chooses an option at random for every question, independently. The number of correct answers is $X$.

   (a)

   Find the probability that the student gets exactly $5$ questions correct.

   [3 marks]

   (b)

   The pass mark is $8$ correct answers. (i) Find the probability that the student passes. (ii) A teacher says that nobody who passes the test can have been choosing at random. Comment on this statement.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A gardener models the number of seeds, $X$, that germinate in a tray as a binomial distribution with mean $6$ and variance $4.2$.

   (a)

   (i) Find the values of $n$ and $p$. (ii) Find $\mathrm{P}(X\le4)$.

   [6 marks]

   (b)

   The gardener records the number of seeds germinating in each of $40$ trays of $20$ seeds. The mean is $6.1$ and the variance is $8.3$. Comment critically on whether a binomial model is appropriate.

   [6 marks]

   Total for question 4: 12 marks

End of questions