Write down the expected number of times that Paula goes to work on a red bus in one week.

In one week, find the probability that she goes to work on a red bus on exactly two days.

In one week, find the probability that she goes to work on a red bus on at least one day.

evidence of binomial distribution (seen anywhere)     (M1)

e.g. \(X \sim {\text{B}}\left( {3{\text{, }}\frac{1}{4}} \right)\)

\({\rm{mean}} = \frac{3}{4}\) (\(= 0.75\))     A1     N2

[2 marks]

\({\rm{P}}(X = 2) = \left( {\begin{array}{*{20}{c}}
3\\
2
\end{array}} \right){\left( {\frac{1}{4}} \right)^2}\left( {\frac{3}{4}} \right)\)     (A1)

\({\rm{P}}(X = 2) = 0.141\) \(\left( { = \frac{9}{{64}}} \right)\)     A1     N2

[2 marks]

evidence of appropriate approach     M1

e.g. complement, \(1 - {\rm{P}}(X = 0)\) , adding probabilities

\({\rm{P}}(X = 0) = {(0.75)^3}\) \(\left( { = 0.422,\frac{{27}}{{64}}} \right)\)     (A1)

\({\rm{P}}(X \ge 1) = 0.578\) \(\left( { = \frac{{37}}{{64}}} \right)\)     A1    N2

[3 marks]

Many candidates did not recognize the binomial nature of this question, suggesting an overall lack of preparation with this topic. Many used 7 days instead of 3 but could still earn marks in follow-through if working was shown. Those who could use their GDC effectively often answered correctly.

Many candidates did not recognize the binomial nature of this question, suggesting an overall lack of preparation with this topic. Many used 7 days instead of 3 but could still earn marks in follow-through if working was shown. Those who could use their GDC effectively often answered correctly.

Many candidates did not recognize the binomial nature of this question, suggesting an overall lack of preparation with this topic. Many used 7 days instead of 3 but could still earn marks in follow-through if working was shown. Those who could use their GDC effectively often answered correctly, although in part (c) some candidates misinterpreted the meaning of “at least one” and found either \({\rm{P}}(X \le 1)\) or \(1 - {\rm{P}}(X \le 1)\) .