Lecture 1 - Sample Spaces

See the [[handout01-ProbBasics-350F24.pdf]]:

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(a)

$S = {Y Y N Y, Y N Y N, N N N N, Y Y Y Y, . . .}$ , all combinations.

We see that $| S | = 2^{4} = 16$ . We can draw a Venn Diagram to show what $S$ is:

(b) It's reasonable since the components don't really depend on each other.

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(d) $B = {Y Y Y Y, Y Y Y N, Y Y N Y, Y N Y Y, N Y Y Y}$ . Here $| B | = 5$ .

(e) A parallel system works iff component 1 works, or component 2 works, or ... for all 4 components. Here component 1 working is event $A_{1}$ , 2 working is $A_{2}$ , and same for 3 and 4.

Thus:

P = A_{1} \cup A_{2} \cup A_{3} \cup A_{4} = ⋃_{i = 1}^{4} A_{i}

Here $| P | = 15$ clearly since it's $P \subseteq S$ while $N N N N \notin P$ is the only element not in $P$ but in $S$ .

(f) Fails would just be the one element $N N N N$ so then $F = {N N N N}$ . Clearly $| F | = 1$ . Alternatively:

F = P^{'} = (A_{1} \cup \dots \cup A_{4})^{'} = A_{1}^{'} \cap A_{2}^{'} \cap A_{3}^{'} \cap A_{4}^{'} $ $ (g) A s e r i e s s y s t e m o n l y w o r k s i f a l l $ A_{1}, . . ., A_{4} $ w o r k, s o :

S_{w} = A_{1}\cap \dots \cap A_

(h) $ S_{f} $ f o r f a i l u r e i s j u s t $ (S_{w})^{'} $ s o :

S_{f}= A_{1}' \cup A_{2}' \cap A_{3}' \cup A_