Homework   1
1.
In   the   design   of   an   electromechanical   product,   seven   different   components   are   to   be   stacked   into   a   cylindrical   casing   that   holds   12   components   in   a   manner   that   minimizes   the   impact   of   shocks.
One   end   of   the   casing   is   designed   as   the   bottom   and   the   other   end   is   the   top.
(a) How   many   different   designs   are   possible?
(b) If   the   seven   components   are   all   identical,   how   many   different   design   are   possible?
(c) If   the   seven   components   consist   of   three   of   one   type   of   component   and   four   of   another   type,   how   many   different   designs   are   possible?
(a)
2.
A   computer   system   uses   passwords   that   are   six   characters   and   each   character   is   one   of   26   letters   (a ‐ z)   or   10   integers   (0 ‐ 9).
Uppercase   letter   are   not   used.
Let   A   denote   the   event   that   a   password   begins   with   a   vowel   (a,e,i,o,u)   and   let   B   denote   the   event   that   password   ends   with   and   event   number   (0,2,4,6,8).
Suppose   a   hacker   selects   a   password   at   random.
Determine   the   following   probabilities:
P(A)      (b)   P(B)    (c)   P(A
∩
B)     (d)   P(A
∪
B)

3.
A   batch   of   500   conta   for   fro zen   orange   ju   contains   5   that   are   de   Two   are   selected,   at   random,   w   repla   from   the   batch.
(a defective?
What   is   the   probability   t   the   seco nd   one   select   is   defectiv   given   that   the   first   one   was
(b
(c   )
What
What
is is
the the
probability probability
t that
both both
e e
defective?
acceptable?
Three   con   are   s elected,   at   ra   witho   replaceme   from   the   batch
(d   What   is   the   probability   t second   on   selected   w   defectiv
the   d   one   selected   is   defective   given   that   th   first   and   he
(e   What   is   the   probability   t   the   third   one   selected   is selected   w   defective   and   the   seco   one   cted   was   okay
defective   given   that   th   first   one
(f f)   What   is   the   probability   t hat   all   three   are   defective
4.
The   pro   that   each   device   functions   is   as   shown.
Ass   that   e   probability   t   each   vice   is   functiona l   does   not   de   on   whe   or   not   o   devices   are   functiona   What   is   th e   probability   that   the   circuit t   operates?

5.
Software   to   detect   fraud   in   customer   phone   cards   tracks   the   number   of   metropolitan   areas   where   calls   originate   each   day.
It   is   found   that   1%   of   the   legitimate   users   originate   calls    from   two   or   more   metropolitan   areas   in   a   single   day.
However,   30%   of   fraudulent   users   originate   calls   from   two   or   more   metropolitan   areas   in   a   single   day.
The   proportion   of   fraudulent   users   is   0.01%.
If   the   same   user   originates   calls   from   two   or   more   metropolitan   areas   in   a   single   day,   what   is   the   probability   that   the   user   is   fraudulent?

Homework   2
1.
Heart   failure   is   due   to   either   natural   occurrences   (87%)   or   outside   factor   (13%).
Outside   factors   are   related   to   induced   substances   or   foreign   objects.
Natural   occurrences   are   caused   by   arterial   blockage,   disease,   and   infection.
Suppose   that   20   patients   will   visit   an   emergency   room   with   heart   failure.
Assume   that   causes   of   heart   failure   between   individuals   are   independent.
(a) What   is   the   probability   that   three   individuals   have   conditions   caused   by   outside   factors?
(b) What   is   the   probability   that   three   or   more   individuals   have   conditions   caused   by   outside   factors?
(c) What   is   the   mean   and   sd.
Of   the   number   of   individuals   with   conditions   caused   by   outside   factors?
2.
A   trading   company   has   eight   computers   that   it   uses   to   trade   on   the   New   York   Stock   Exchange
(NYSE).
The   probability   of   a   computer   failing   in   a   day   is   0.005
and   the   computer   fail   independently.
Computers   are   repaired   in   the   evening   and   each   day   is   an   independent   trial.
(a) What   is   the   probability   that   all   eight   computers   fail   in   a   day?
(b) What   is   the   mean   number   of   a   days   until   a   specific   computer   fails?
(c) What   is   the   mean   number   of   a   days   until   all   eight   computer   fails   in   the   same   day?

3.The
number   of   flaws   in   bolts   of   cloth   in   textile   manufacturing   is   assumed   to   be   Poisson   distributed   with   a   mean   of   0.1
flaw   per   square   meter.
(a) What   is   the   probability   that   there   are   two   flaws   in   1   square   meter   of   cloth?
(b) What   is   the   probability   that   there   is   one   flaw   in   10   square   meters   of   cloth?
(c) What   is   the   probability   that   there   is   no   flaws   in   20   square   meters   of   cloth?
(d) What   is   the   probability   that   there   are   at   least   2   flaws   in   10   square   meters   of   cloth?
4.
A   congested   computer   network   has   a   1%   chance   of   losing   a   data   packet   and   packet   losses   are   independent   events.
An   email   message   requires   100   packets.
(a) What   is   the   distribution   of   data   packets   that   must   be   resent?
Include   the   parameter   values.
(b) What   is   the   probability   that   two   or   more   packets   must   be   sent?
(c) What   is   the   mean   and   sd.
of   the   number   of   packet   that   must   be   resent?
(d) If   there   are   10   messages   and   each   contains   100   packets,   what   is   the   probability   that   at   least   one   message   requires   two   or   more   packets   be   resent?

Homework   3
1.
The   compressive   strength   of   samples   of   cement   can   be   modeled   by   a   normal   distribution   with   a   mean   of   6000   Kg/cm
2
and   sd.
of   100   Kg/cm
2
a) What   is   the   probability   that   a   sample’s   strength   is   less   than   6250   Kg/cm
2
?
b) What   is   the   probability   that   a   sample’s   strength   is   between   5800   and   5900   Kg/cm
2
?
c) What   strength   is   exceeded   by   95%   of   the   samples?
2.
The   manufacturing   of   semiconductor   chips   produces   2%   of   defective   chips.
Assume   the   chips   are   independent   and   that   a   lot   contains   1000   chips.
a) Approximate   the   probability   that   more   than   25   chips   are   defective?
b) Approximate   the   probability   that   between   20   and   30   chips   are   defective?

3.
The   time   between   arrivals   of   small   aircraft   at   a   county   airport   is   exponentially   distributed   with   a   mean   of   one   hour.
a) What   is   the   probability   that   more   than   three   aircraft   arrive   within   an   hour?
b) If   30   separate   one ‐ hour   intervals   are   chosen,   what   is   the   probability   that   no   interval   contains   more   than   three   arrivals?
c) Determine   the   length   of   an   interval   of   time(in   hours)   such   that   the   probability   that   no   arrivals   occur   during   the   interval   is   0.10