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Two forecasts · Part 1 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.

Assume the rain events are independent. What is the probability of rain in at least one market?

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

0.580.58

Independence makes the overlap 0.30 times 0.40, or 0.12. Inclusion-exclusion then gives 0.58. Adding 0.30 and 0.40 directly would count days when both markets are wet twice. The complement gives the same answer by another route: both markets stay dry with probability 0.70 times 0.60, or 0.42, so at least one is wet with probability 0.58.

Worked solution

  1. Formula
    P(A∪B)=P(A)+P(B)−P(A)P(B)P(A\cup B)=P(A)+P(B)-P(A)P(B)
  2. Substitute
    0.30+0.40−0.30(0.40)0.30+0.40-0.30(0.40)
  3. Solve
    0.70−0.12=0.580.70-0.12=0.58
  4. Answer
    0.580.58

Sanity check. The union must be at least 0.40 and at most 0.70.

Takeaway: Independence determines the overlap; it does not remove it.

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

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Two forecasts, part 1 of 3 · QuantMax