Aussie Maths & Science Teachers: Save your time with SmarterEd
In a chocolate factory the material for making each chocolate is sent to a machines.
The time, `X` seconds, taken to produce a chocolate by machine is a binomial distribution where it can be shown that `P(X <= 3) = 9/32`.
A random sample of 10 chocolates is chosen. Find the probability, correct to two decimal places, that exactly 4 of these 10 chocolates took 3 or less seconds to produce. (2 marks)
`0.18`
`text(Let)\ \ C = \ text(number of chocolates that take less than 3 seconds)`
`C ∼\ text(Bin)(n, p) ∼\ text(Bin)(10, 9/32)`
| `P(C = 4)` | `=((10),(4)) (9/32)^4 (23/32)^6` |
| `=0.181…` | |
| `=0.18\ \ \ text{(2 d.p.)}` |
Shoddy Ltd produces statues that are classified as Superior or Regular and are entirely made by machines, on a construction line. The quality of any one of Shoddy’s statues is independent of the quality of any of the others on its construction line. The probability that any one of Shoddy’s statues is Regular is 0.8.
Shoddy Ltd wants to ensure that the probability that it produces at least one Superior statues in a day’s production run is at least 0.95.
Calculate the minimum number of statues that Shoddy would need to produce in a day to achieve this aim. (3 marks)
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`14`
`text(Let)\ \ X = text(Number of superior statues)`
`X∼\ text(Bin) (n, 0.2)`
| `P(X >= 1)` | `>= 0.95` |
| `1-P(X = 0)` | `>= 0.95` |
| `1-((n),(0)) (0.8^n) (0.2^0)` | `>= 0.95` |
| `1-0.8^n` | `>=0.95` |
| `0.8^n` | `<=0.05` |
| `n xx ln 0.8` | `<=ln 0.05` |
| `n` | `>= ln 0.05/ln 0.8,\ \ \ text{(ln 0.8 < 0)}` |
| `n` | `>= 13.4…` |
| `:. n_min` | `= 14` |