1. Suppose a coin is tossed three times.
   1. Find a sample space for this experiment.
   2. Find the event E that at least one toss is a head.
2. Suppose a fuse is tested until it fails.
   1. Find a sample space for this experiment.
   2. Let $E_k$ be the event that a fuse fails in less than $k$ hours, $k = 1, 2, \ldots$. Find
      1. $E_1 \cup E_2 \cup E_3$
      2. $E_1 \cap E_2 \cap E_3$
      3. $E_3 \cap E_2^c$
      4. $\cup_{k=1}^\infty E_k$
      5. $\cap_{k=1}^\infty E_k$

Answers:

1. a. $\Omega = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$

b. $E = \{HHH, HHT, HTH, THH, HTT, THT, TTH\}$

2. a. $\Omega = [0, \infty)$

b. I. $E_1 \cup E_2 \cup E_3 = [0, 3)$

II. $E_1 \cap E_2 \cap E_3 = [0, 1)$

III. $E_3 \cap E_2^c = [2, 3)$

IV. $\cup_{k=1}^\infty E_k = \Omega = [0, \infty)$

V. $\cap_{k=1}^\infty E_k = [0, 1)$