The binomial distributionEdexcel A-Level Maths: Subtopic test

1. 1
   A student guesses the answer to every question on a 10-question multiple-choice test. Each question has four options, exactly one of which is correct. The random variable $X$ is the number of questions answered correctly, and $X\sim B(10,0.25)$.

   (a)

   Find $P(X=3)$.

   [1 mark]

   • A$0.0021$
   • B$0.224$
   • C$0.250$
   • D$0.776$

   (b)

   Find the probability that the student gets at least one question right.

   [1 mark]

   • A$0.0563$
   • B$0.944$
   • C$0.750$
   • D$0.250$

   (c)

   Find $P(X\ge4)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A gardener plants 12 seeds. Each seed germinates with probability $0.9$, independently of the others. The random variable $Y$ is the number of seeds that germinate.

   (a)

   Find the probability that all 12 seeds germinate.

   [1 mark]

   • A$0.282$
   • B$0.718$
   • C$0.377$
   • D$0.900$

   (b)

   Find $P(Y\le10)$.

   [1 mark]

   • A$0.282$
   • B$0.659$
   • C$0.230$
   • D$0.341$

   (c)

   Find $P(Y=10)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A machine produces bolts. Each bolt is defective with probability $0.05$, independently of the others. Bolts are packed in samples of 20, and $X$ is the number of defective bolts in a sample.

   (a)

   State the distribution of $X$ and find the probability that a sample contains exactly 2 defective bolts.

   [3 marks]

   (b)

   A sample is accepted if it has at most 2 defective bolts. Five independent samples are inspected. Find the probability that exactly 4 of them are accepted.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A basketball player scores with each free throw with probability $0.6$, independently of the other throws. She takes 12 free throws, and $X$ is the number of throws she scores.

   (a)

   (i) Find $P(X=7)$. (ii) Find $P(X\ge9)$. (iii) Find the largest integer $k$ such that $P(X\ge k)>0.9$.

   [6 marks]

   (b)

   She takes $n$ free throws. Find the least value of $n$ for which the probability that she scores at least once is greater than $0.99$.

   [6 marks]

   Total for question 4: 12 marks

End of questions