Best bound on a dry weekend without independence

Question

Saturday's rain probability is $40\%$ and Sunday's is $70\%$ . Making no assumption about how the two days are related, what is the smallest possible upper bound on the probability that neither day sees rain?

Short answer

$0.3$ . A dry weekend is the complement of the union of the two rain events. The union is at least as likely as its larger piece, $0.7$ , with equality when rain on Saturday forces rain on Sunday, so the dry probability can be as high as $1-0.7=0.3$ and no higher.

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Lessons behind this question

Counting and Combinatorics
Independence
Indeterminate Forms and L'Hôpital's Rule

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