Chapter 23

The probability that a component fails within a month is \(0.009\). If 800 components are examined calculate the probability that the number failing within a month is (a) nine, (b) five, (c) less than three, (d) four or more.

A fair die is thrown three times. Calculate the probability of obtaining (a) three \(6 \mathrm{~s}\) (b) three \(1 \mathrm{~s}\) (c) two \(6 \mathrm{~s}\) given the first number is a 1 (d) two 6 s given the first number is a 6 .

The temperature, \(T{ }^{\circ} \mathrm{C}\), of a freezer follows a normal distribution with mean \(-6{ }^{\circ} \mathrm{C}\) and standard deviation of \(2{ }^{\circ} \mathrm{C} .\) Calculate the probability that (a) \(T>-5\) (b) \(T<-7\) (c) \(-6

Silicon chips are manufactured by four machines, A, B, C and D. Machines A, B, C and D manufacture \(20 \%, 25 \%, 35 \%\) and \(20 \%\) of the components respectively. Of those silicon chips manufactured by machine A, \(2.1 \%\) are faulty. The respective figures for machines \(\mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\) are \(3 \%, 1.6 \%\) and \(2.5 \%\). A silicon chip is selected at random. Calculate the probability that it is (a) made by machine \(\mathrm{C}\) and is faulty (b) made by machine \(\mathrm{A}\) and is not faulty (c) faulty.

Give two examples of (a) discrete data, (b) continuous data.

The probability that a machine has a lifespan of more than 7 years is \(0.85\). Twelve machines are chosen at random. Calculate the probability that (a) 10 have a lifespan of more than 7 years (b) 11 have a lifespan of more than 7 years (c) 10 or more have a lifespan of more than 7 years.

Find the variance and standard deviation of the following frequency distribution: $$ \begin{array}{rr} \hline x & f \\ \hline 6 & 7 \\ 7 & 3 \\ 8 & 2 \\ 9 & 4 \\ 10 & 2 \\ \hline \end{array} $$

Classify the following variables as discrete or continuous: (a) the number of times a machine breaks down in 12 months (b) the time between breakdowns of a machine (c) the capacitance of a capacitor (d) the amount of money in your pocket (e) the number of hairs on your head.

A machine manufactures 350 micro-chips per hour. The probability that a chip is faulty is \(0.012\). Calculate the probability that in a particular hour there are (a) one, (b) three, (c) more than three faulty chips manufactured.

Components are made by machines A, B and C. Machine A makes \(35 \%\) of the components, machine B makes \(25 \%\) and machine C makes the rest. Two components are picked at random. Calculate the probability that (a) both are made by machine \(\mathrm{C}\) (b) one is made by machine A and one is made by machine B (c) exactly one is made by machine \(\mathrm{A}\) (d) at least one is made by machine B (e) both are made by the same machine.