6 Two bags contain coloured discs. At first, bag \(P\) contains 2 red discs and 2 green discs, and bag \(Q\) contains 3 red discs and 1 green disc. A disc is chosen at random from bag \(P\), its colour is noted and it is placed in bag \(Q\). A disc is then chosen at random from bag \(Q\), its colour is noted and it is placed in bag \(P\). A disc is then chosen at random from bag \(P\).
The tree diagram shows the different combinations of three coloured discs chosen.
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- Write down the values of \(a , b , c , d , e\) and \(f\).
The total number of red discs chosen, out of 3, is denoted by \(R\). The table shows the probability distribution of \(R\).
| \(r\) | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( R = r )\) | \(\frac { 1 } { 10 }\) | \(k\) | \(\frac { 9 } { 20 }\) | \(\frac { 1 } { 5 }\) |
- Show how to obtain the value \(\mathrm { P } ( R = 2 ) = \frac { 9 } { 20 }\).
- Find the value of \(k\).
- Calculate the mean and variance of \(R\).