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Statistics, EXT1 S1 EQ-Bank 12

A manufacturer of pool cue tips knows that 4% of pool cue tips produced in its factory need to be scrapped.

A random sample of 10 cue tips produced in the factory is examined.

  1. What is the probability that exactly 3 cue tips need to be scrapped?

     

    Give your answer correct to 3 decimal places.  (1 mark)

  2. What is the probability that, at most, 2 cue tips need to be scrapped?

     

    Give your answer correct to 3 decimal places.  (2 marks)

Show Answers Only
  1. `0.006`
  2. `0.993`
Show Worked Solution

i.     `text(Let)\ \ X = text(number of defective cue tips)`

`X\ ~\ text(Bi) (10, 0.04)`

`P(X = 3)` `= \ ^10 C_3 ⋅ (0.04)^3 (0.96)^7`
  `= 0.00577…`
  `=0.006\ \ text{(to 3 d.p.)}`

 

ii.   `P(X <= 2)` `= P(X = 0) + P(X = 1) + P(X = 2)`
    `=\ ^10 C_0(0.04)^0 (0.96)^10 + \ ^10 C_1(0.04) (0.96)^9 + \ ^10 C_2 (0.04)^2 (0.96)^8`
    `= 0.66483 + 0.27701 + 0.05194`
    `= 0.994\ \ text{(to 3 d.p.)}`

Statistics, EXT1 S1 SM-Bank 3

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)

Show Answers Only

`0.18`

Show Worked Solution

`text(Let)\ \ C = \ text(number of chocolates that take less than 3 seconds)`

COMMENT: Take care as  `P(X <= 3) = 9/32`  provides the equivalent of  `p` here.

`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.)}`

Statistics, EXT1 S1 SM-Bank 1

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)

Show Answers Only

`14`

Show Worked Solution

`text(Let)\ \ X = text(Number of superior statues)`

♦♦ Mean mark 27%.

`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 ln 0.8` `<=ln 0.05`
`n` `>= ln 0.05/ln 0.8,\ \ \ text{(ln 0.8 < 0)}`
`n` `>= 13.4…`
`:. n_min` `= 14`