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Rumor Mill (Posted on 2004-12-03) Difficulty: 4 of 5
Waldo is having a party and has 50 guests, among whom is his brother Basil.

Basil starts a rumor about Waldo; a person hearing this rumor for the first time will then tell another person chosen uniformly at random the rumor, with the exceptions that no one will tell the rumor to Waldo or to the person they heard it from.

If a person who already knows the rumor hears it again, they will not tell it again.

What's the probability that everyone, except Waldo, will hear the rumor before it stops propagating?

What if each person told two people chosen uniformly at random?

No Solution Yet Submitted by SilverKnight    
Rating: 4.0000 (6 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Better Simulation for Part 2 | Comment 5 of 13 |
(In reply to Stat. analysis of Charlie's part 2 by Jer)

The problem with the simulation was that it used QuickBasic's built-in random number generator, which starts repeating every so often.   The program calls it millions of times, so it is cycling over the random numbers more than once, and it apparently was repeating a smaller number of trials several times, rather than a million independent trials.

So I modified the program to use binary random numbers downloaded from www.random.org, and, out of one million trials, 343 trials resulted in all 50 guests knowing the rumor.  That's a probability of .000343, or 1 in 2915.  The 343 is quite close to the expected 339.8....

The revised program is

defdbl a-z
FOR trial = 1 TO 1000000
  REDIM person(50)
  REDIM who(50)
  person(1) = 1: p1Ct = 1: p2Ct = 0
  DO
    FOR i = 1 TO 50
      IF person(i) = 1 THEN
       DO
        r = rand50(1)
       LOOP UNTIL r <> i AND r <> who(i)
       DO
        s = rand50(1)
       LOOP UNTIL s <> i AND s <> who(i) AND s <> r
       IF person(r) = 0 THEN
         person(r) = 1
         p1Ct = p1Ct + 1
         who(r) = i
       END IF
       IF person(s) = 0 THEN
         person(s) = 1
         p1Ct = p1Ct + 1
         who(s) = i
       END IF
       person(i) = 2
       p2Ct = p2Ct + 1
       p1Ct = p1Ct - 1
      END IF
    NEXT
  LOOP UNTIL p1Ct = 0
  IF p2Ct = 50 THEN suc = suc + 1
  tot = tot + 1
  IF trial MOD 100 = 0 THEN
   PRINT suc, tot
  END IF
NEXT

FUNCTION rand50 (x)
 STATIC started
 STATIC number
 STATIC pulled
 IF started = 0 THEN OPEN "160megs.bin" FOR BINARY AS #1: started = 1: pulled = 3
 IF pulled > 2 THEN
   s$ = "   "
   GET #1, , s$
   IF EOF(1) THEN STOP
   number = ASC(s$) + ASC(MID$(s$, 2)) * 256# + ASC(MID$(s$, 3)) * 256# * 256#
   pulled = 0
 END IF
 r = number MOD 50 + 1
 number = number \ 50
 pulled = pulled + 1
 rand50 = r
END FUNCTION

 


  Posted by Charlie on 2004-12-03 20:53:37
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