IB Maths AI HL AI 4.3 Probability and distributions questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1 Paper 2

Difficulty

Easy Medium Hard

Question 1

Paper 2Easy5 marks

A school surveys 50 students about how they travelled to school on a particular day. Of these students, 23 travelled by bus, 19 travelled by train, and 8 used both the bus and the train.

One student is selected at random from the 50 students.

(a)

Find the probability that the student travelled by bus or train.

[2]

(b)

Find the probability that the student travelled by neither bus nor train.

[1]

(c)

Given that the student travelled by bus, find the probability that the student also travelled by train.

[2]

Question 2

Paper 2Medium5 marks

At a parcel centre, $X$ is the excess handling time, in minutes, for a randomly selected parcel. Figure 1 shows the probability distribution of $X$

(a)

Calculate $\mathrm{E}(X)$ and $\mathrm{Var}(X)$

[3]

(b)

For a parcel with excess handling time $X$ the additional handling charge $Y$ in AUD, is modelled by $Y=3X+2$ Hence determine $\mathrm{E}(Y)$ and $\mathrm{Var}(Y)$

[2]

A local emergency department records two types of admissions during each 20-minute interval. The two streams of admissions are independent, and admissions occur independently at a uniform average rate throughout each interval. The mean numbers are 1.4 type A admissions and 0.9 type B admissions per interval.

A planner is considering a normal, binomial or Poisson model for the total number, $N$ of admissions in an interval.

(a)

Show that a suitable model for $N$ is a Poisson distribution with mean 2.3.

[2]

(b)

The department can assess at most $C$ admissions during an interval. The planner requires the probability that the department can assess all admissions to be at least 0.95.

Using your model from part (a), determine the smallest possible integer value of $C$ Show numerical evidence that it is the smallest value satisfying the requirement.

[4]

Question 4

Paper 2Hard6 marks

A snack company models the mass, $X$ g, of a packet as normally distributed with mean $503$ g and standard deviation $4.8$ g. Packets with mass below $494$ g do not meet the lower specification. The company wants at least $94\%$ of packets to meet the specification $494\leq X\leq U$ where $U$ is the upper specification.

Use your GDC for normal probability and inverse normal calculations.

(a)

Show that $P(X<494)=0.0304$

[2]

(b)

Using your result from part (a), determine the smallest value of $U$ to the nearest $0.1$ g, that meets the company’s requirement. A manager proposes setting $U$ to the 95th percentile of the distribution. Assess whether this proposal meets the requirement.

[4]

The best questions are still waiting

Unlock every question in this topic, then type, photograph or draw your answer and have it marked against the markscheme. Find out where you are dropping marks.

Unlock all questions Compare Student and Student Plus