Represent this information on a Venn Diagram.

There were \(28\) students who used a bus to travel to school. Calculate the number of students
(i)     who travelled by car and by bus but did not walk;
(ii)    who travelled by car.

Tomoko used a bus to travel to school yesterday.

Find the probability that she also walked.

Two students are chosen at random from all \(50\) students.

Find the probability that
(i)     both students walked;
(ii)    only one of the students walked.

     (A4)
Note: Award (A1) for rectangle and three labelled intersecting circles, (A1) for \(3\), (A1) for \(5\) and \(10\), (A1) for \(7\) and \(12\).

(i)     \(28 - (10 + 3 + 7) = 8\)     (M1)(A1)(ft)(G2)

Note: Follow through from their Venn diagram.

(ii)    \(5 + 3 + 8 + 12 = 28\)     (M1)(A1)(ft)(G2)

Note: Follow through from part (b)(i) and their Venn diagram.

\({\text{P(}}\left. {{\text{walk}}} \right|{\text{bus}}) = \frac{{13}}{{28}}\) \((0.464{\text{, }}46.4\% )\) (\(0.464285 \ldots \))     (A1)(A1)(ft)(G2)

Note: Award (A1)(ft) for the numerator, (A1) for denominator.

(i)     \(\frac{{23}}{{50}} \times \frac{{22}}{{49}}\)     (A1)(M1)(M1)

Note: Award (A1) for \(23\) seen, (M1) for non replacement, (M1) for multiplying their fractions.

\( = \frac{{506}}{{2450}}\) \((0.207{\text{, }}20.7\% )\) (\(0.206530 \ldots \))     (A1)(G3)

(ii)    \(\frac{{23}}{{50}} \times \frac{{27}}{{49}} + \frac{{27}}{{50}} \times \frac{{23}}{{49}}\)     (A1)(ft)(M1)

Notes: Award (A1)(ft) for two products, (M1) for adding two products. Do not penalise in (ii) for consistent use of with replacement.

\( = \frac{{1242}}{{2450}}\) \((0.507{\text{, }}50.7\% )\) (\(0.509638 \ldots \))     (A1)(ft)(G2)