AppliedNumeric answerFree solution

Two forecasts · Part 2 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.

Drop the independence assumption. What is the largest possible 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

0.700.70

The largest union occurs when the two rain events never coincide. Then their overlap is zero and the union is 0.70. This is possible because their probabilities sum to less than one. Inclusion-exclusion shows why: the union is 0.70 minus the overlap, so every unit of overlap is a unit of union lost. The opposite extreme, rain in A only ever on days when B is wet too, gives the smallest union of 0.40, and the independent answer of 0.58 sits between the two.

Worked solution

  1. Formula
    P(A∪B)=P(A)+P(B)−P(A∩B)P(A\cup B)=P(A)+P(B)-P(A\cap B)
  2. Substitute
    0.30+0.40−00.30+0.40-0
  3. Solve
    0.700.70
  4. Answer
    0.700.70

Sanity check. A probability cannot exceed one; here the sum is only 0.70.

Takeaway: When dependence is unknown, make the overlap as small as feasible for the largest union.

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

More probability questions

Two forecasts, part 2 of 3 · QuantMax