2022 DHS P2 Q7

Commodity \(X\) is traded four times a week from Monday to Thursday. The unit price of \(X\) can only rise or fall on any day. If the unit price of \(X\) rises on a day, there is a probability of \(0.6\) that it will rise on the next trading day. If the unit price of \(X\) falls on a day, there is a probability of \(0.15\) that it will rise on the next trading day.

In a particular week, the unit price of \(X\) rises on Monday and the events \(A\) and \(B\) are defined as follows.

\(A:\) the unit price of \(X\) falls on Tuesday.

\(\;B:\) the unit price of \(X\) rises on Thursday.

1. Find
   1. \(\mathrm{P}\left( A\cap B \right)\),[2]
   2. \(\mathrm{P}\left( B \right)\),[2]
   3. \(\mathrm{P}\left( \left. B \right|A \right)\).[2]
2. State, with a reason, whether \(A\) and \(B\) are independent.[1]
3. If the unit price of \(X\) rises for \(12\) consecutive Mondays in a particular year, find the probability that it rises on exactly \(5\) of the corresponding Tuesdays.[2]