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I am trying to get to grips with the Bayes' formula by developing an intuition about the formula itself, and on how to use it, and how to interpret.

Please, take a problem, and my questions written within them - I will highlight my questions and will post them as I add the information.

**The problem:**

A corporation plans to announce an expansion into overseas markets. The expansion will occur only if the overseas demand is sufficient to support the necessary sales. And if demand is sufficient and overseas expansion occurs, the Corporation is likely to raise its prices.

O represents the event of overseas expansion; I represents a price increase; I

^{C}represents no price increase.

The unconditional and conditional probabilities are estimated as follows:

P(I) = 0.3

P(I

^{C)}= 0.7

P(O | I) = 0.6

P(O | I

^{C}) = 0.4

**question1:**Do I understand correctly that by giving P(I) and P(I

^{C)}we assume that the corporation might increase/not increase its prices in any case, independently of any changes in the business model, i.e. independently whether it expanses overseas, or it doesn't?

**question2:**If the first assumption is correct, then we must determine probabilities that the corporation might, or might not, expand its business overseas in case it first raises its prices (or doesn't raise them), correct?

Therefore, that should become our base case probability assumptions.

The problem asks to solve P(I | O) - the probability of price increase if the corporation announces the overseas expansion.

Then the text book proceeds in the following way.

Given the multiplication rule and the joint probability rule:

P(O | I) = P(IO) / P(I) and P(IO) = P( I | O) x P(O)

hence, from the second formula P(I | O) = P(IO) / P(O)

and from the first formula P(IO) = P(O | I) x P(I)

=> P(I | O) = [P(O | I) x P(I)] / P(O)

Using the total probability rule we determine P(O) = P(O | I) x P(I) + P(O | I

^{C}) x P(I

^{C)}) =

0.6 x 0.3 + 0.4 x 0.7 = 0.46

=> P(I | O) = [0.6 x 0.3] / 0.46 = 0.3913

=> if the new information expand overseas is announced, then the prior probability estimate of P(I) = 0.30 should be increased to 0.3913.

The problem is solved.

**Question3:**It all sounds so artificial to me, and I have a trouble understanding why we are using such approach. To find the probability of "increased prices given the expansion overseas" we first shall estimate the probability of increased prices on its own, independently of any event; then estimate if any expansion can occur in case prices are increased prior to the expansion; and then based on these estimates find the probability of a reverse situation - increased prices given the expansion occurred.

Sounds so bewildering.

I will be grateful for your help.

Thank you so much.