6 The proportions of different colours of loom bands in a box of 10000 loom bands are given in the table.
| Colour | Blue | Green | Red | Orange | Yellow | White |
| Proportion | 0.25 | 0.25 | 0.18 | 0.12 | 0.15 | 0.05 |
- A sample of 50 loom bands is selected at random from the box.
Use a binomial distribution with \(n = 50\), together with relevant information from the table, to estimate the probability that this sample contains:
- exactly 4 red loom bands;
- at most 10 yellow loom bands;
- at least 30 blue or green loom bands;
- more than 35 but fewer than 45 loom bands that are neither yellow nor white.
- The random variable \(R\) denotes the number of red loom bands in a random sample of \(\mathbf { 3 0 0 }\) loom bands selected from the box.
Estimate values for the mean and the variance of \(R\).
[0pt]
[2 marks]