- (a) Briefly explain what is meant by a sample space.
(b) State two properties which a function \(f ( x )\) must have to be a probability function. - A company makes two cars, model \(A\) and model \(B\). The distance that model \(A\) travels on 10 litres of petrol is normally distributed with mean 109 km and variance \(72.25 \mathrm {~km} ^ { 2 }\). The distance that model \(B\) travels on 10 litres of petrol is normally distributed with mean 108.5 km and variance \(169 \mathrm {~km} ^ { 2 }\).
In a trial, one of each model is filled with 10 litres of petrol and sent on a journey of 110 km . Find which model has the greater probability of completing this journey, and state the value of this probability. - \(A , B\) and \(C\) are three events such that \(\mathrm { P } ( A ) = x , \mathrm { P } ( B ) = y\) and \(\mathrm { P } ( C ) = x + y\).
It is known that \(\mathrm { P } ( A \cup B ) = 0.6\) and \(\mathrm { P } ( B \mid A ) = 0.2\).
(a) Show that \(4 x + 5 y = 3\).
It is also known that \(B\) and \(C\) are mutually exclusive and that \(\mathrm { P } ( B \cup C ) = 0.9\)
(b) Obtain another equation in \(x\) and \(y\) and hence find the values of \(x\) and \(y\).
(c) Deduce whether or not \(A\) and \(B\) are independent events.