2012 RJC P2 Q10

In the gold market, there are \(4\) trading days, Monday to Thursday, in a week. The gold price can only rise or fall on any day. If the gold price rises on a day, there is a probability of \(0.7\) that it will rise on the next day; and if the gold price falls on a day, there is a probability of \(0.1\) that it will rise on the subsequent day.

In a particular week, the gold price falls on the Monday, the events \(A\) and \(B\) are defined as follows.

\(A\): the gold price rises on Tuesday.

\(B\): the gold price falls on Thursday.

Find

1. \(\mathrm{P}\left( A\cap B \right)\),[2]
2. \(\mathrm{P}\left( B \right)\),[3]
3. \(\mathrm{P}\left( A'\cup B \right)\),[2]
4. \(\mathrm{P}\left( B|A \right)\).[2]

State, with a reason, whether \(A\) and \(B\) are independent.[1]