The binomial distributionAQA A-Level Maths: Subtopic test

1. 1
   A gardener plants 10 seeds. Each seed germinates with probability $0.3$, independently of the others. The number of seeds that germinate is $X$, where $X\sim B(10,0.3)$.

   (a)

   Find $P(X=3)$.

   [1 mark]

   • A$0.0022$
   • B$0.267$
   • C$0.3$
   • D$0.650$

   (b)

   Find $P(X\ge4)$.

   [1 mark]

   • A$0.650$
   • B$0.200$
   • C$0.850$
   • D$0.350$

   (c)

   State two conditions that must hold for $X$ to be modelled by a binomial distribution in this context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A multiple-choice test has 20 questions, each with four options of which exactly one is correct. A student guesses every answer. The number of correct answers is $X$, where $X\sim B(20,0.25)$.

   (a)

   Find $P(X=5)$.

   [1 mark]

   • A$0.202$
   • B$0.617$
   • C$0.415$
   • D$0.25$

   (b)

   Find $P(X\le4)$.

   [1 mark]

   • A$0.190$
   • B$0.585$
   • C$0.415$
   • D$0.617$

   (c)

   Find the probability that the student gets more than 7 questions correct.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A factory makes light bulbs and 8% of them are defective, independently of one another. A box contains 25 bulbs. The number of defective bulbs in a box is $Y$, where $Y\sim B(25,0.08)$.

   (a)

   Find the probability that a box contains exactly 2 defective bulbs.

   [3 marks]

   (b)

   A box is rejected if it contains more than 3 defective bulbs. Find the probability that a box is rejected. A retailer receives 5 boxes. Find the probability that at least one of them is rejected.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A clinic books 12 appointments for a morning session. Each patient independently fails to attend with probability $0.15$. The number of patients who do not attend is $X$, where $X\sim B(12,0.15)$.

   (a)

   Find (i) $P(X=2)$, (ii) $P(X\ge4)$, (iii) $P(2\le X\le4)$.

   [6 marks]

   (b)

   The clinic has only 10 appointment slots in the session. Find the probability that more patients attend than there are slots. Over 5 independent sessions, find the probability that this happens at least once. State one assumption that may not be realistic in this context.

   [6 marks]

   Total for question 4: 12 marks

End of questions