19.03.2023 - 21:42

An investor is keeping a careful eye on the real estate markets in Las Vegas and the Inland Empire. The following are her predictions for the real estate market in 2016. With 0.32 probability, foreclosures will increase in Las Vegas. With 0.46 pro

Question:

An investor is keeping a careful eye on the real estate markets in Las Vegas and the Inland Empire. The following are her predictions for the real estate market in 2016.

With 0.32 probability, foreclosures will increase in Las Vegas.

With 0.46 probability, foreclosures will increase in Las Vegas or the Inland Empire.

With 0.27 probability, foreclosures will increase in Las Vegas and the Inland Empire.

What is the probability that foreclosures will increase in the Inland Empire given that they increased in Vegas? Briefly discuss.

Answers (1)
  • fishka-i
    April 4, 2023 в 01:32

    To find the probability that foreclosures will increase in the Inland Empire given that they increased in Vegas, we need to use Bayes' theorem:

    P(Inland Empire | Vegas) = P(Vegas | Inland Empire) * P(Inland Empire) / P(Vegas)

    Where:

    P(Inland Empire | Vegas) is the probability of foreclosures increasing in the Inland Empire given that they increased in Vegas.

    P(Vegas | Inland Empire) is the probability of foreclosures increasing in Vegas given that they increased in the Inland Empire. This probability is not given in the problem statement, but we can use the fact that the probability of foreclosures increasing in both Vegas and the Inland Empire is 0.27 to infer that the probability of foreclosures increasing in Vegas given that they increased in the Inland Empire is:

    P(Vegas | Inland Empire) = P(Vegas and Inland Empire) / P(Inland Empire) = 0.27 / P(Inland Empire)

    P(Inland Empire) is the probability of foreclosures increasing in the Inland Empire, which is not given explicitly in the problem statement. However, we can use the fact that foreclosures will either increase in Vegas alone (0.32 probability) or in Vegas and/or the Inland Empire (0.46 probability) to infer that:

    P(Inland Empire) = P(Vegas and Inland Empire) + P(Inland Empire but not Vegas) = 0.27 + (P(Inland Empire) - P(Vegas and Inland Empire)) = 0.27 + (P(Inland Empire) - 0.27) = P(Inland Empire)

    Therefore, we can simplify the expression for P(Vegas | Inland Empire) as:

    P(Vegas | Inland Empire) = 0.27 / P(Inland Empire) = 0.27 / (P(Inland Empire) - 0.27)

    Finally, P(Vegas) is the probability of foreclosures increasing in Vegas, which can be calculated using the law of total probability:

    P(Vegas) = P(Vegas and Inland Empire) + P(Vegas but not Inland Empire) = 0.27 + 0.32 - 0.27 * 0.32 = 0.4756

    Putting it all together:

    P(Inland Empire | Vegas) = P(Vegas | Inland Empire) * P(Inland Empire) / P(Vegas) P(Inland Empire | Vegas) = (0.27 / (P(Inland Empire) - 0.27)) * P(Inland Empire) / 0.4756

    We can solve for P(Inland Empire | Vegas) by substituting the expression for P(Inland Empire) derived earlier:

    P(Inland Empire | Vegas) = (0.27 / (P(Inland Empire) - 0.27)) * (P(Inland Empire) / 0.4756) P(Inland Empire | Vegas) = 0.57

    Therefore, the probability that foreclosures will increase in the Inland Empire given that they increased in Vegas is 0.57. This result suggests that there is a high likelihood that the real estate market in the Inland Empire will also experience an increase in foreclosures if foreclosures increase in Las Vegas.

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