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Two forecasts · Part 3 of 3

A desk cares whether either of two markets has a weather disruption tomorrow. The chance of rain in market A is 0.30; in market B it is 0.40.

You now learn that both markets are wet on 0.20 of days. Conditional on rain in market B, what is the chance of rain in market A?

Answer with a number. Fractions, powers and expressions like 23/6 or C(52,5) are read correctly in practice.

Show the answer and worked solution

Answer

0.500.50

Conditioning changes the denominator to days when B is wet. Half of those days also have rain in A. The original independence model would have implied 0.30 instead, so the new observation changes the forecast. A common slip is to divide by P(A) instead, giving 0.20/0.30, about 0.67; that is the chance of rain in B given rain in A, which answers a different question.

Worked solution

  1. Formula
    P(A∣B)=P(A∩B)P(B)P(A\mid B)=\frac{P(A\cap B)}{P(B)}
  2. Substitute
    0.20/0.400.20/0.40
  3. Solve
    0.500.50
  4. Answer
    0.500.50

Sanity check. The result exceeds the unconditional 0.30, consistent with positive association.

Takeaway: A measured overlap replaces the independence assumption; condition on the event you are given.

Answer it in practice – your answer is marked and recorded.

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Two forecasts, part 3 of 3 · QuantMax