I am solving the following decision problem. I formulated the problem into the tree OK but am unsure of what the probabilities along the branches are. i.e. how do I go from, for example, $P(H_1 \cap H_2)$ to $P(H_2)$?
Tom has 100, 000 savings and he uses 50, 000 to buy a house
The probability of having highly dangerous fires in year 1 and moderately dangerous ones in year 2 in the area where the house resides is equal to 0.25. The probability of having moderately dangerous fires in year 1 and highly dangerous ones in year 2 is equal to 0.15. The probability of having highly dangerous fires in year 1 and year 2 is equal to 0.2. The probability of having moderately dangerous fires in year 1 and year 2 is equal to 0.4.
Let $H_k$ be the event of having a highly dangerous fire in year $k$ and $M_K$ be the event of having a moderately dangerous fire in year in $k$.
So we have: $P(H_1 \cap M_2)=0.25$, $P(M_1 \cap H_2)=0.15$, $P(H_1 \cap H_2)=0.2$ and $P(M_1 \cap M_2)=0.4$
If a highly dangerous fire happens during the first year, then Tom insures the house for year 2. In any case, and independently of the previous events, Tom insures the house for year 3.
Insurance costs start at 500 for year 1. In each year this rises by 200 if there was a highly dangerous fire the year before. If a moderate fire, it reduces by 100.
Draw and solve the decision tree , maximising Tom's residual savings
Here is my decision tree:
Hopefully I have the nodes and insurance costs (written in red) correct. Although I am unsure how to calculate the probabilities along the branches. How do we use the given probabilities of the form $P(A \cap B)$ to find $P(A)$ etc?
Thank you for your help