\(180\) people were interviewed and asked what types of transport they had used in the last year from a choice of airplane \((A)\), train \((T)\) or bus \((B)\). The following information was obtained.

\(47\) had travelled by airplane

\(68\) had travelled by train

\(122\) had travelled by bus

\(25\) had travelled by airplane and train

\(32\) had travelled by airplane and bus

\(35\) had travelled by train and bus

\(20\) had travelled by all three types of transport

Draw a Venn diagram to show the above information.

Find the number of people who, in the last year, had travelled by

(i)      bus only;

(ii)     both airplane and bus but not by train;

(iii)    at least two types of transport;

(iv)    none of the three types of transport.

A person is selected at random from those who were interviewed.

Find the probability that the person had used only one type of transport in the last year.

Given that the person had used only one type of transport in the last year, find the probability that the person had travelled by airplane.

(A1)(A1)(A1)(A1)

Note: Award (A1) for a rectangle with \(3\) intersecting clearly labelled circles.
Award (A1) for \(20\) in correct region.
Award (A1) for \(15\), \(12\), \(5\) in correct regions.
Award (A1) for \(75\), \(28\), \(10\) in correct regions.

(i)      \(75\)            (A1)(ft)

Note: Follow through from their Venn diagram.

(ii)     \(12\)            (A1)(ft)

Note: Follow through from their Venn diagram.

(iii)    \(15 + 20 + 12 + 5\)            (M1)

\( = 52\)            (A1)(ft)(G2)

Note: Award (M1) for addition of their \(15\), \(20\), \(12\) and \(5\). Follow through from their Venn diagram.

(iv)    \(180 - 165\)            (M1)

Note: Award BI for their \(165\), or a sum adding to their \(165\), seen.

\(15\)            (A1)(ft)(G2)

Note: Follow through from their Venn diagram.

\(\frac{{113}}{{180}}\,\,\,(0.628,\,\,62.8\,\% ,\,\,0.62777...)\)         (A1)(ft)(A1)(G2)

Note: Award (A1)(ft) for correct numerator. Follow through from their Venn diagram.
Award (A1) for \(180\) in the denominator.

\(\frac{{10}}{{113}}\,\,\,(0.0885,\,\,8.85\,\% ,\,\,0.08849...)\)          (A1)(ft)(A1)(ft)(G2)

Note: Award (A1)(ft) for correct numerator, (A1)(ft) for correct denominator. Follow through from their Venn diagram or numerator from part (c).

Question 1: Sets and probability
In part (a), a surprising number of candidates could not construct the Venn diagram correctly, based on the given information. This led to problems with the rest of the parts although they were usually awarded follow-through marks in part (b). Part (b) which required interpreting the information from their Venn diagram was generally well done. Some candidates gave the probability rather than number of people. Most candidates were successful at the simple probability but many struggled with the conditional probability.

Question 1: Sets and probability
In part (a), a surprising number of candidates could not construct the Venn diagram correctly, based on the given information. This led to problems with the rest of the parts although they were usually awarded follow-through marks in part (b). Part (b) which required interpreting the information from their Venn diagram was generally well done. Some candidates gave the probability rather than number of people. Most candidates were successful at the simple probability but many struggled with the conditional probability.

Question 1: Sets and probability
In part (a), a surprising number of candidates could not construct the Venn diagram correctly, based on the given information. This led to problems with the rest of the parts although they were usually awarded follow-through marks in part (b). Part (b) which required interpreting the information from their Venn diagram was generally well done. Some candidates gave the probability rather than number of people. Most candidates were successful at the simple probability but many struggled with the conditional probability.

Question 1: Sets and probability
In part (a), a surprising number of candidates could not construct the Venn diagram correctly, based on the given information. This led to problems with the rest of the parts although they were usually awarded follow-through marks in part (b). Part (b) which required interpreting the information from their Venn diagram was generally well done. Some candidates gave the probability rather than number of people. Most candidates were successful at the simple probability but many struggled with the conditional probability.