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math-340:m340-f15-hw:hw-1 [2015/08/25 08:02] – created dcsmath-340:m340-f15-hw:hw-1 [2015/08/31 05:41] (current) dcs
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   - Suppose a fuse is tested until it fails.   - Suppose a fuse is tested until it fails.
     - Find a sample space for this experiment.     - Find a sample space for this experiment.
-    - Let $E_k$ be the event that a fuse fails in less than $k$ hours, $k = 1, 2, \ldots$. Find $E_1 \cup E_2 \cup E_3$$E_1 \cap E_2 \cap E_3$ $\cup_{k=1}^\infty E_k$, and $\cap_{k=1}^\infty E_k$+    - Let $E_k$ be the event that a fuse fails in less than $k$ hours, $k = 1, 2, \ldots$. Find  
 +      - $E_1 \cup E_2 \cup E_3$ 
 +      - $E_1 \cap E_2 \cap E_3$ 
 +      - $E_3 \cap E_2^c$ 
 +      - $\cup_{k=1}^\infty E_k$ 
 +      - $\cap_{k=1}^\infty E_k$ 
 + 
 + 
 +Answers: 
 + 
 +1. a. $\Omega = \{HHHHHT, HTH, THH, HTT, THT, TTH, TTT\}$ 
 + 
 +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. $\cap_{k=1}^\infty E_k = [0, 1)$