S2 June 2015 Q7

Notes

B1: using \(\mathrm{P}(10) = 0.2\ (p)\), \(\mathrm{P}(20) = 0.4\ (q)\) and \(\mathrm{P}(50) = 0.4\ (r)\) may be seen in calculations or implied by a correct probability.

B1: three correct medians and no extras.

P(Median 10): M1: allow if \((p + q + r) = 1\) and use \(p^3 + 3\times p^2\times q + 3\times p^2\times r\) or \(p^3 + 3\times p^2\times(q + r)\) look for \(\dfrac{1}{125} + \dfrac{6}{125} + \dfrac{6}{125}\)

P(Median 50): M1: allow if \((p + q + r) = 1\) and use \(r^3 + 3\times r^2\times p + 3\times r^2\times q\) or \(r^3 + 3\times r^2\times(p + q)\) Look for \(\dfrac{8}{125} + \dfrac{12}{125} + \dfrac{24}{125}\)

P(Median 20): M1: allow if \((p + q + r) = 1\) and use \(3\times p\times q^2 + 6\times p\times q\times r + q^3 + 3\times q^2\times r\) \(\dfrac{12}{125} + \dfrac{24}{125} + \dfrac{8}{125} + \dfrac{24}{125}\)

How to award the M marks – Allow the use of 1, 2 and 5 for the medians for the method marks
M1 any correct calculation (implied by correct answer) for P(m = 10) or P(m = 20) or P(m = 50)
M1 any 2 correct calculations (implied by 2 correct answers) P(m = 10) or P(m = 20) or P(m = 50)
M1 any 3 correct calculations (implied by 3 correct answers) for P(m = 10) and P(m = 20) and P(m = 50) or 3 probabilities that add up to 1 providing it is 1 – their 2 other calculated probabilities. Do not allow \(\dfrac{1}{5}\ \dfrac{2}{5}\ \dfrac{2}{5}\)
NB if they do not have a correct answer their working must be clear including the addition signs.

A1: awrt any 1 correct
A2: awrt all 3 correct
These do not need to be in a table as long as the correct probablity is with the correct median(10, 20 & 50)
NB: Do Not allow the use of 1,2 and 5 for the medians for the A marks