4.8 Binomial distributionIB Maths: Applications and Interpretation SL: Subtopic test

1. 1
   A multiple-choice quiz has 12 questions, each with four options of which exactly one is correct. A student guesses every answer at random. Let $X$ be the number of questions answered correctly.

   (a)

   Which of the following gives the distribution of $X$?

   [1 mark]

   • A$\mathrm{B}(12,0.5)$
   • B$\mathrm{B}(12,0.75)$
   • C$\mathrm{B}(4,0.25)$
   • D$\mathrm{B}(12,0.25)$

   (b)

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

   [1 mark]

   • A$0.25$
   • B$3$
   • C$2.25$
   • D$9$

   (c)

   Use your GDC to find the probability that the student answers exactly 3 questions correctly.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A basketball player scores a free throw with probability 0.7, independently each time. She takes 8 free throws. Let $Y$ be the number of free throws she scores.

   (a)

   Find the probability that she scores all 8 throws.

   [1 mark]

   • A$0.0576$
   • B$0.942$
   • C$6.56\times10^{-5}$
   • D$5.6$

   (b)

   Use your GDC to find the probability that she scores at least 7 throws.

   [1 mark]

   • A$0.198$
   • B$0.0576$
   • C$0.255$
   • D$0.552$

   (c)

   Find the mean and the variance of $Y$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A factory makes light bulbs. Each bulb is defective with probability 0.04, independently of the others. A box contains 25 bulbs. Let $D$ be the number of defective bulbs in a box.

   (a)

   (i) Write down the distribution of $D$.
   (ii) State two conditions that must hold for this model to be appropriate.

   [3 marks]

   (b)

   Use your GDC to find
   (i) the probability that no bulb in the box is defective;
   (ii) the probability that at least two bulbs in the box are defective.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A call centre finds that 35% of its calls result in a sale, independently of each other. An employee makes 15 calls in one hour. Let $S$ be the number of calls in that hour that result in a sale.

   (a)

   (i) Write down the distribution of $S$.
   (ii) Use your GDC to find $P(S=5)$.
   (iii) Use your GDC to find the probability that the employee makes at least 8 sales.
   (iv) Find the expected number of sales in the hour.

   [6 marks]

   (b)

   (i) Use your GDC to find the probability that the employee makes between 3 and 7 sales inclusive.
   (ii) Find the smallest value of $k$ such that $P(S\leq k)>0.9$.
   (iii) The employee makes 15 calls in each of five separate hours, independently. Find the probability that in exactly two of these hours the employee makes at least 8 sales.

   [6 marks]

   Total for question 4: 12 marks

End of questions