Question

An economist estimates a 65% probability that the economy will expand next year. The technology sector has a 70% probability of outperforming the market if the economy expands and a 20% probability of outperforming the market if the economy does not expand. Given the new information that the technology sector will not outperform the market, what is the probability that the economy will not expand?

Answer 1

We are given here that:
P( economy expand ) = 0.65, therefore P(economy dont expand ) = 1 - 0.65 = 0.35

Also, we are given here that: technology sector has a 70% probability of outperforming the market if the economy expands and a 20% probability of outperforming the market if the economy does not expand, therefore we get here:

P( tech outperforms | economy expands) = 0.7 and
P( tech outperforms | economy dont expand) = 0.2

Using law of total probability, we get here:
P(tech outperforms) = P( tech outperforms | economy expands)P(economy expands) + P( tech outperforms | economy dont expand)P(economy dont expand)

P(tech outperforms) = 0.7*0.65 + 0.2*0.35 = 0.525

Using Bayes theorem, we get here:
P( economy dont expand | tech outperforms) = P( tech outperforms | economy dont expand)P(economy dont expand) / P( tech outperforms ) = 0.2*0.35 / 0.525 = 0.1333

Therefore 0.1333 is the required probability here.