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Solution proposal assignment 7:
A quick intro to probability notation:
Pr(x|i) means the probability of x given i.
Pr(x & i) means the probability of x and i occuring together.
Pr(x) means the unconditional probability of x occuring.

Now let's solve the problem at hand:

Apriori:
Pr(g|H) = 0.6
Pr(b|H) = 0.4
Pr(g|L) = 0.1
Pr(b|L) = 0.9

Needed for later:
Pr(H & g) = Pr(H) * Pr(g|H) = 0.4 * 0.6 = 0.24
Pr(H & b) = Pr(H) * Pr(b|H) = 0.4 * 0.4 = 0.16
Pr(L & g) = Pr(L) * Pr(g|L) = 0.6 * 0.1 = 0.06
Pr(L & b) = Pr(L) * Pr(b|L) = 0.6 * 0.9 = 0.54
Pr(g) = Pr(H & g) + Pr(L & g) = 0.30
Pr(b) = Pr(H & b) + Pr(L & b) = 0.70

Aposteriori probabilities:
Pr(H|g) = Pr(H & g)/Pr(g) = 0.24/0.30 = 0.8
Pr(H|b) = Pr(H & b)/Pr(b) = 0.16/0.70 ~= 0.229
Pr(L|g) = Pr(L & g)/Pr(g) = 0.06/0.30 = 0.2
Pr(L|b) = Pr(L & b)/Pr(b) = 0.54/0.70 ~= 0.771

Optimal choices based on expected profits:
Apriori:
Marketing the product yields expected profits: 0.4 * 35 000 + 0.6 * 5 000 = 17 000
Not marketing yields: 15 000
Thus you choose to market apriori.

Aposteriori:
Expected profits if indication is g:
If marketing: Pr(H|g) * 35 000 + Pr(L|g) * 10 000 = 29 000
If not marketing: 15 000
You choose marketing.

Expected profits if indication is b:
If marketing: Pr(H|b) * 35 000 + Pr(L|b) * 15 000 = 11 870
If not marketing: 15 000
You choose not marketing.

Value of testing = Pr(g) * 29 000 + Pr(b) * 15 000 - 17 000 = 2 200

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