6.
A spinner has edges numbered \(1,2,3,4\) and 5 . When the spinner is spun, the number of the edge on which it lands is the score. The probability distribution of the score, \(N\), is given in the table.
| Score, \(N\) | 1 | 2 | 3 | 4 | 5 |
| Probability | 0.3 | 0.2 | 0.2 | \(x\) | \(y\) |
It is known that \(\mathrm { E } ( N ) = 2.55\).
- Find \(\operatorname { Var } ( N )\).
- Find \(\mathrm { E } ( 3 N + 2 )\).
- Find \(\operatorname { Var } ( 3 N + 2 )\).
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A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.
(a) Find the probability of exactly 4 faults in a 15 metre length of cloth.
(b) Find the probability of more than 10 faults in 60 metres of cloth.
A retailer buys a large amount of this cloth and sells it in pieces of length \(x\) metres. He chooses \(x\) so that the probability of no faults in a piece is 0.80
(c) Write down an equation for \(x\) and show that \(x = 1.7\) to 2 significant figures.
The retailer sells 1200 of these pieces of cloth. He makes a profit of 60 p on each piece of cloth that does not contain a fault but a loss of \(\pounds 1.50\) on any pieces that do contain faults.
(d) Find the retailer's expected profit.
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