Friday, December 16, 2011

segments to form a polygon


What is the probability that a stick of length 1 divided randomly into 3 parts can form a triangle with the three parts?

& then, General Bonus asks u...Soldier, what's the probability that a polygon with (N+1) sides can be made from (N+1) segments obtained by randomly dividing a stick of length l into (N+1) parts?

You gave me the probability formula . Here’s my explanation --
Suppose we mark a circle’s 12 o’clock position with “0”, 3 o’clock with 90, 6 o’clock with 180 etc. So each position on the circle has a unique number from 0 to 359.999999999.

We put N+1 random dots (all at once) on the circle’s circumference, creating N+1 segments. (If any segment exceeds half the circle’s circumference, then we would fail to form a polygon.)

Rotate the clock to align 12 o’clock position to any random dot. Break the circle at that dot (and losing that dot). Now we have N dots and N+1 segments.

Now we color the segments. Say black segment is first from the 12 o’clock position, 2nd is red segment.

P (black segment exceeding half the circle) = P (all N dots landing beyond 6 o’clock position) = 2-N

P (red segment exceeding half the circle) ? Now keep all the dots and the colors. Rejoin the circle. Rotate it by the length of black segment. Break circle there. Now red segment is the new 1st segment. P (red segment exceeding half) = 2-N

There are N+1 segments, so P (any segment exceeding half the circle) = (N+1)*2-N

No comments:

Total Pageviews

my favorite topics (labels)

_fuxi (302) _misLabel (13) _orig? (3) _rm (2) _vague (2) clarified (58) cpp (39) cpp_const (22) cpp_real (76) cpp/java/c# (101) cppBig4 (54) cppSmartPtr (35) cppSTL (33) cppSTL_itr (27) cppSTL_real (26) cppTemplate (28) creditMkt (14) db (65) db_sybase (43) deepUnder (31) dotnet (20) ECN (27) econ/bank` (36) fin/sys_misc (43) finGreek (34) finReal (45) finRisk (30) finTechDesign (46) finTechMisc (32) finVol (66) FixedIncom (28) fMath (7) fMathOption (33) fMathStoch (67) forex (39) gr8IV_Q (46) GTD_skill (15) GUI_event (30) inMemDB (42) intuit_math (41) intuitFinance (57) javaMisc (68) javaServerSide (13) lambda/delegate (22) marketData (28) math (10) mathStat (55) memIssue (8) memMgmt (66) metaProgram` (6) OO_Design (84) original_content (749) polymorphic/vptr (40) productive (21) ptr/ref (48) py (28) reflect (8) script`/unix (82) socket/stream (39) subquery/join (30) subvert (13) swing/wpf (9) sysProgram` (16) thread (164) thread_CAS (15) thread_cpp (28) Thread* (22) timeSaver (80) transactional (23) tune (24) tuneDB (40) tuneLatency (30) z_ajax (9) z_algoDataStruct (41) z_arch (26) z_arch_job (27) z_automateTest (17) z_autoTrad` (19) z_bestPractice (39) z_bold (83) z_bondMath (35) z_book (18) z_boost (19) z_byRef^Val (32) z_c#GUI (43) z_c#misc (80) z_cast/convert (28) z_container (67) z_cStr/arr (39) z_Favorite* (8) z_FIX (15) z_forex (48) z_fwd_Deal (18) z_gz=job (33) z_gzBig20 (13) z_gzMgr (13) z_gzPain (20) z_gzThreat (19) z_hib (19) z_IDE (52) z_ikm (5) z_IR_misc (36) z_IRS (26) z_javaWeb (28) z_jdbc (10) z_jobFinTech (46) z_jobHunt (20) z_jobRealXp (10) z_jobStrength (15) z_jobUS^asia (27) z_letter (42) z_linq (10) z_memberHid` (11) z_MOM (54) z_nestedClass (5) z_oq (24) z_PCP (12) z_pearl (1) z_php (20) z_prodSupport (7) z_py (31) z_quant (14) z_regex (8) z_rv (38) z_skillist (48) z_slic`Problem (6) z_SOA (14) z_spring (25) z_src_code (8) z_swingMisc (50) z_swingTable (26) z_unpublish (2) z_VBA/Excel (8) z_windoz (17) z_wpfCommand (9)

About Me

New York (Time Square), NY, United States
http://www.linkedin.com/in/tanbin