Umbrella sales depend on the weather

Question

There is a 30\% chance of rain tomorrow. On a rainy day your shop sells 50 umbrellas on average, and on a dry day it sells 10 umbrellas on average. How many umbrellas should you expect to sell tomorrow?

Accepted Answer

Two branches, rain or dry. The law of total expectation (tower rule) averages the two branch outcomes, weighted by how likely each is: E[	ext{sales}] = E\big[\,E[	ext{sales} \mid 	ext{weather}]\,\big]. Step 1, the branch averages are given: rainy day 50, dry day 10. Step 2, weight by the weather odds, 30\% rain and 70\% dry: E[	ext{sales}] = 0.30 	imes 50 + 0.70 	imes 10 = 15 + 7 = 22. You should expect to sell 22 umbrellas. Rainy days sell far more, but dry days are more common, so the blend lands closer to the dry-day figure. Takeaway. Weight each scenario's average by the chance it h…

Umbrella sales depend on the weather

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