1. 1
   A seed company states that each of its tomato seeds germinates with probability 0.85, independently of the other seeds. A packet contains 20 seeds. Let $X$ be the number of seeds in the packet that germinate, so that $X\sim B(20,\,0.85)$. A calculator may be used in this question.

   (a)

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

   [1 mark]

   • A$3$
   • B$2.55$
   • C$17$
   • D$1.60$

   (b)

   Find the standard deviation of $X$, correct to 3 significant figures.

   [1 mark]

   • A$2.55$
   • B$0.357$
   • C$4.12$
   • D$1.60$

   (c)

   Find the probability that exactly 17 seeds in the packet germinate.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A quiz has 12 multiple-choice questions. Each question has four options, exactly one of which is correct. A student who has not revised guesses the answer to every question at random, independently. Let $X$ be the number of questions the student answers correctly. A calculator may be used in this question.

   (a)

   Find the probability that the student answers every question incorrectly, correct to 3 significant figures.

   [1 mark]

   • A$0.0317$
   • B$5.96\times10^{-8}$
   • C$0.968$
   • D$0.75$

   (b)

   Find the most likely number of correct answers.

   [1 mark]

   • A$2$
   • B$3$
   • C$4$
   • D$9$

   (c)

   The pass mark is 6 correct answers out of 12. Find the probability that the student passes.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A basketball player scores each free throw she attempts with probability 0.78. The outcomes of her free throws are independent of each other. A calculator may be used in this question.

   (a)

   In one match she attempts 15 free throws. Find the probability that she scores at least 12 of them.

   [3 marks]

   (b)

   Find the least number of free throws she must attempt so that the probability that she misses at least one of them is greater than 0.99.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A flight has 180 seats. The airline sells $n$ tickets for the flight, where $n>180$. Each ticket holder turns up for the flight with probability 0.95, independently of all other ticket holders. The flight is overbooked if more than 180 ticket holders turn up. Let $X$ be the number of ticket holders who turn up. A calculator may be used in this question.

   (a)

   (i) Give one reason why the assumption of independence may not be realistic in this context.
   (ii) For $n=186$, find the mean and the standard deviation of $X$.
   (iii) For $n=186$, find the probability that the flight is overbooked, and interpret your answer in context.

   [6 marks]

   (b)

   The airline decides that the probability that the flight is overbooked must be less than 0.05.
   (i) Find the greatest number of tickets it can sell.
   (ii) The airline sells this number of tickets for the flight on each of 7 days. The days are independent. Find the probability that the flight is overbooked on at least one of the 7 days.

   [6 marks]

   Total for question 4: 12 marks

End of questions