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1.4 Exercise 20

Avg difficulty 2.5

Use fundamental properties of probability to show

  1. P ( A B ) ≤ P ( A ) ≤ P ( A ∪ B ) ≤ P ( A ) + P ( B ) P \left(A B\right) \leq P \left(A\right) \leq P \left(A \cup B\right) \leq P \left(A\right) + P \left(B\right)
  2. P ⋂ j = 1 ∞ E j ≤ P ( E i ) ≤ P ⋃ j = 1 ∞ E j ≤ ∑ j = 1 ∞ P ( E j ) P \left(\cap_{j = 1}^{\infty}E_{j}\right) \leq P \left(E_{i}\right) \leq P \left(\cup_{j = 1}^{\infty}E_{j}\right) \leq \sum_{j = 1}^{\infty} P \left(E_{j}\right)

Pfeiffer, Applied Probability, 1.4 Problems on Probability Systems, exercise 20

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Adapted from Applied Probability © 2009 Paul E. Pfeiffer. Licensed under CC BY 3.0. Source: Applied Probability. Imported at revision d7a0420.