TITLE: Probability and Interest Rates of Markets
QUESTION [1 upvotes]: Q: Suppose the current interest rate in Dollars is at 6% while those
  in Pounds and in Yen are at 5%. Suppose that it is thought in the next
  6 months the interest rates for Dollars will stay the same with
  probability 0.7, go up by 1% with probability 0.1 and go down by 1%
  with probability 0.2; likewise for Pounds and likewise for Yen. Also
  it is known that changes in these interest rates in the next six
  months are independent of each other.

Find the probability that in six months' time the Dollar interest rate is still greater than either Pound or Yen interest rate.
[For these types of questions I use modelling with probability trees and values at different stages of time...but for this one I'm not sure how to model the rates can someone start me in the right direction it would be much appreciated thank you!]

REPLY [1 votes]: A table could be useful:
$\begin{array}{|c|c|c|c|} \hline & 0.7&0.1&0.2  \\ \hline \$ & 6 &7&5 \\ \hline ￥&5 &6&4 \\ \hline £ &5 &6&4 \\ \hline \end{array}$
The probability, that Dollar interest rate is smaller or equal than the Pound interest rate and the Yen interest rate, given the Dollar interest rate is 6%, is $0.1\cdot 0.1=0.1^2$
The probability, that Dollar interest rate is smaller or equal than the Pound interest rate and the Yen interest rate, given the Dollar interest rate is 7%, is $0\%$
The probability, that Dollar interest rate is smaller or equal than the Pound interest rate and the Yen interest rate, given the Dollar interest rate is 5%, is $(0.7+0.1)\cdot (0.7+0.1)=0.8^2$
Now use the Law of total probability to calculate the probability, that the dollar interest rate is smaller or equal than the Pound interest rate and the Yen interest rate.
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