Statistical Measures

2018/10/06 · by · in Fun, Math

In the stats book that I used at college, A First Course in Probability (sixth ed) by Sheldon Ross, I found two problems that seem paradoxical when juxtaposed. Can you explain the opposite results?

Ch 2 Axioms of Probability, Self-Test Exercise #15.

Show that if P(A_i) = 1 for all i\geq1, then P\left(\bigcap\limits_{i=1}^{\infty} A_i\right) = 1.

Ch 5 Continuous Random Variables, Theoretical Exercise #6.

Define a collection of events E_a, 0 < a < 1, having the property that P(E_a) = 1 for all a, but P\left(\bigcap\limits_{a} E_a\right) = 0.
Hint: Let random variable X be uniform over (0,1) and define E_a in terms of X.