91Ó°ÊÓ

Components are made by machines \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\). Machine A makes \(17 \%\) of the components, machine B makes \(21 \%\) of the components, machine \(\mathrm{C}\) makes \(20 \%\) of the components and machine D makes the remainder. For machine A, \(96 \%\) of the components are reliable, for machine B, \(89 \%\) are reliable, for machine C, \(92 \%\) are reliable and for machine D, \(97 \%\) are reliable. A component is picked at random. Calculate the probability that it is (a) reliable (b) not reliable (c) reliable, given it is made by machine B (d) not reliable, given it is made by machine \(\mathrm{D}\) (e) made by machine A given it is reliable (f) made by machine \(\mathrm{C}\) given it is unreliable

Capacitors are manufactured by four machines, \(1,2,3\) and 4. The probability a capacitor is manufactured acceptably varies according to the machine. The probabilities are \(0.94,0.91,0.97\) and \(0.94\), respectively, for machines \(1,2,3\) and 4 . (a) A capacitor is taken from each machine. What is the probability all four capacitors are acceptable? (b) Two capacitors are taken from machine 1 and two from machine \(2 .\) What is the probability all four capacitors are acceptable? (c) A capacitor is taken from each machine. Calculate the probability that at least three capacitors are acceptable. (d) A capacitor is taken from each machine. From this sample of four capacitors, one is taken at random. (i) What is the probability it is acceptable and made by machine 1 ? (ii) What is the probability it is acceptable and made by machine \(2 ?\) (e) A capacitor is taken from each machine. From this sample of four capacitors, one is taken at random. What is the probability it is acceptable? [Hint: use the results in (d).]

Circuit boards are made by machines A, B, C and D. Machine A makes \(15 \%\) of the components, machine B makes \(30 \%\), machine \(\mathrm{C}\) makes \(35 \%\) and machine \(\mathrm{D}\) makes the remainder. The probability that a board is acceptable is \(0.93\) when made by machine \(\mathrm{A}, 0.96\) when made by machine B, \(0.95\) when made by machine \(\mathrm{C}\) and \(0.93\) when made by machine D. A board is picked at random. Calculate the probability that it is (a) made by machine D (b) made by machine A and is acceptable (c) made by machine B and is unacceptable (d) made by machine \(\mathrm{C}\) and is acceptable (e) made by machine \(\mathrm{D}\) and is unacceptable (f) unacceptable and made by machine \(\mathrm{C}\) (g) In a batch of 1000 boards, how many would be expected to be acceptable and made by machine D?