# Statistical Measures

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 $X$.