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I added som explanatory notes:

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:
A priori probabilities are the probabilities we have at hand before we conduct the PPC testing (from which we get a indication/response/signal/message):
Pr(g|H) = 0.6
Pr(b|H) = 0.4
Pr(g|L) = 0.1
Pr(b|L) = 0.9

So, e.g. Pr(g|H) is the probability of a good signal, given that the demand is high.

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:
These are the probabilities for a state of demand, given a signal response from the PPC testing. E.g., Pr(H|g) is the probability of high demand, given a good signal (g).
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:
In this stage we figure out what our optimal choices are (to market or not to market), under our different possible situations. First we find out what our optimal responses are before we have a signal from testing, i.e. a priori.
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, because it has higher expected profits.

Aposteriori:
This is where we check what actions are the most profitable if we conduct PPC testing and receive the different signals (good or bad indication).
Expected profits if indication is g:
If we choose marketing after a good signal from testing: 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 after a bad indication: Pr(H|b) * 35 000 + Pr(L|b) * 15 000 = 11 870
If not marketing: 15 000
You choose not marketing.

Value of testing = What you are willing to spend to get the result = Difference in expected profits a posteriori and a priori = Probability of a good signal * Expected profits from our optimal choice given that signal + Probability of a bad signal * Expected profits from our optimal choice given that signal - expected profits from optimal choice a priori = Pr(g) * 29 000 + Pr(b) * 15 000 - 17 000 = 2 200

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