6\% of the counters labelled C are red
One counter is selected at random from the bag.
- Complete the tree diagram on the opposite page to illustrate this information.
- Calculate the probability that the counter is labelled A and is not red.
- Calculate the probability that the counter is red.
- Given that the counter is red, find the probability that it is labelled C
\end{enumerate}
\includegraphics[max width=\textwidth, alt={}, center]{b8ac20db-4237-4def-81aa-a3eecbeefbdd-15_1155_1000_285_456}
5. A discrete random variable \(Y\) has probability function
$$\mathrm { P } ( \mathrm { Y } = \mathrm { y } ) = \left\{ \begin{array} { c l }
\mathrm { k } ( 3 - \mathrm { y } ) & y = 1,2
\mathrm { k } \left( \mathrm { y } ^ { 2 } - 8 \right) & y = 3,4,5
\mathrm { k } & y = 6
0 & \text { otherwise }
\end{array} \right.$$
where \(k\) is a constant. - Show that \(k = \frac { 1 } { 30 }\)
Find the exact value of
- \(\mathrm { P } ( 1 < Y \leqslant 4 )\)
- \(\mathrm { E } ( Y )\)
The random variable \(X = 15 - 2 Y\)
- Calculate \(\mathrm { P } ( Y \geqslant X )\)
- Calculate \(\operatorname { Var } ( X )\)
- Three events \(A , B\) and \(C\) are such that
$$\mathrm { P } ( A ) = 0.1 \quad \mathrm { P } ( B \mid A ) = 0.3 \quad \mathrm { P } ( A \cup B ) = 0.25 \quad \mathrm { P } ( C ) = 0.5$$
Given that \(A\) and \(C\) are mutually exclusive - find \(\mathrm { P } ( A \cup C )\)
- Show that \(\mathrm { P } ( B ) = 0.18\)
Given also that \(B\) and \(C\) are independent,
- draw a Venn diagram to represent the events \(A , B\) and \(C\) and the probabilities associated with each region.