4.17 Poisson distributionIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   Calls to a school office arrive independently at a uniform average rate of $4$ calls per hour. Let $N$ be the number of calls in a one-hour period.

   (a)

   Assuming $N$ has a Poisson distribution, use your GDC to find $\mathrm{P}(N=3)$.

   [1 mark]

   • A$0.238$
   • B$0.433$
   • C$0.762$
   • D$0.195$

   (b)

   Find the variance of the number of calls in a three-hour period.

   [1 mark]

   • A$4$
   • B$36$
   • C$12$
   • D$3.46$

   (c)

   Find the probability that at least $2$ calls arrive in a $30$-minute period.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Flaws occur in rolls of fabric independently of each other, at a uniform average rate of $0.8$ per square metre in fabric X and $1.5$ per square metre in fabric Y.

   (a)

   One square metre of fabric X and one square metre of fabric Y are inspected. What is the distribution of the total number of flaws found?

   [1 mark]

   • A$\mathrm{Po}(2.3)$
   • B$\mathrm{Po}(1.15)$
   • C$\mathrm{Po}(1.2)$
   • D$\mathrm{Po}(0.7)$

   (b)

   Use your GDC to find the probability that no flaws are found in the two square metres described in (a).

   [1 mark]

   • A$0.449$
   • B$0.100$
   • C$0.223$
   • D$0.900$

   (c)

   Two square metres of fabric X and one square metre of fabric Y are inspected. Use your GDC to find the probability that more than $3$ flaws are found in total.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Patients arrive at the emergency department of a hospital during the night at an average rate of $5$ per hour. Let $X$ be the number of patients who arrive in a one-hour period during the night.

   (a)

   State two conditions needed for $X$ to be modelled by a Poisson distribution, and give one reason why arrivals at an emergency department might not satisfy a condition.

   [3 marks]

   (b)

   Assume that $X$ has a Poisson distribution.
   (i) Find the probability that exactly $7$ patients arrive in a one-hour period.
   (ii) Find the probability that more than $12$ patients arrive in a two-hour period.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A factory makes glass bottles on two production lines. On each line a bottle is cracked with probability $0.03$, independently of other bottles. Air bubbles occur in the glass independently, at a uniform average rate of $0.5$ per bottle on line 1 and $0.8$ per bottle on line 2.

   (a)

   Eighty bottles from line 1 are chosen at random.
   (i) State a suitable distribution for the number of cracked bottles $C$, giving its parameters and one reason it is suitable.
   (ii) Find $\mathrm{P}(C\ge4)$.
   (iii) Three bottles from line 1 are chosen at random. Find the probability that they contain exactly $2$ bubbles in total.

   [6 marks]

   (b)

   One bottle is chosen from line 1 and one from line 2. Let $X_1$ and $X_2$ be the numbers of bubbles in these two bottles and let $T=X_1+X_2$.
   (i) State the distribution of $T$.
   (ii) Find $\mathrm{P}(T\ge3)$.
   (iii) Find the probability that the bottle from line 1 has no bubbles and that $T=2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions