Each day the exchange rate of the pound against the euro either rises or falls.

Suppose that observation has shown that

$-$if the exchange rate rises on a given day, then there is a $40\%$

probability that it will rise again the next day, and otherwise it will fall;

$-$if the exchange rate falls on a given day, then there is a $65\%$

probability that it will fall again the next day, and otherwise it will rise.

$($a$)$ Draw a state diagram to represent this system.

$[5$ marks$]$

$($b$)$ Write down the transition matrix for this system.

$[3$ marks$]$

$($c$)$ If the exchange rate rises on Monday, find the probability that it rises

$($i$)$ on Tuesday;

$[2$ marks$]$

$($ii$)$ on Thursday.

$[4$ marks$]$

$($d$)$ Find the proportion of days on which the exchange rate rises in the long term.

$[7$ marks$]$

$($e$)$ Over a long period, would you expect the exchange rate to rise or fall

overall? Give one reason why we cannot be certain about this behaviour.

$[4$ marks$]$