Find the value of \(k\).

Write down \({\text{P}}(X = 2)\).

Find \({\text{P}}(X = 2|X > 0)\).

valid approach     (M1)

eg\(\,\,\,\,\,\)total probability = 1

correct equation     (A1)

eg\(\,\,\,\,\,\)\(0.475 + 2{k^2} + \frac{k}{{10}} + 6{k^2} = 1,{\text{ }}8{k^2} + 0.1k - 0.525 = 0\)

\(k = 0.25\)     A2     N3

[4 marks]

\({\text{P}}(X = 2) = 0.025\)     A1     N1

[1 mark]

valid approach for finding \({\text{P}}(X > 0)\)     (M1)

eg\(\,\,\,\,\,\)\(1 - 0.475,{\text{ }}2({0.25^2}) + 0.025 + 6({0.25^2}),{\text{ }}1 - {\text{P}}(X = 0),{\text{ }}2{k^2} + \frac{k}{{10}} + 6{k^2}\)

correct substitution into formula for conditional probability     (A1)

eg\(\,\,\,\,\,\)\(\frac{{0.025}}{{1 - 0.475}},{\text{ }}\frac{{0.025}}{{0.525}}\)

0.0476190

\({\text{P}}(X = 2|X > 0) = \frac{1}{{21}}\) (exact), 0.0476     A1     N2

[3 marks]