SOLUTIONS AND PROBLEMS TO PREVIOUS POSTING:
Resampling, (bootstrapping, etc.), is revolutionizing the practice of
statistic, and the way it is taught. For articles, and information on
software + books, results of classroom trails, and descriptions of pending
projects in which teachers can become involved, contact the University of
Maryland's Resampling Project. We are especially interested in class
testers. Contact:
Resampling Project,
attn. P.G. Bruce
College of Business
University of Maryland,
College Park, MD 20742.
Phone: 703-522-2713
FAX 703-522-5846
email pcbruce at wam.umd.edu
(mark attn. P. C. Bruce).
please provide both postal and email address.
SOLUTIONS:
Solution, Puzzle 1:
1. Three urns - "0,0"; "0,1"; "1,1".
2. Choose an urn at random. [Can do this on the computer
with random selection of urn numbers, and then "If"]
3. Choose the first element in the chosen urn's vector. If
"1", stop trial and make no further record. If "0", continue.
4. Record the second element in the chosen urn's vector on
the scoreboard.
5. Repeat (2 - 5), and calculate the proportion "0s" on
scoreboard. (Answer should be 2/3.)
NUMBERS (1 2 3) a '"1" denotes a selection of urn 1 (u1),
'"2" urn 2, "3" urn 3
NUMBERS (6 6) u1 '"u1" has two pennies (6's)
NUMBERS (7 7) u2 '"u2" has two nickels (7's)
NUMBERS (6 7) u3 '"u3" has one penny & one nickel
REPEAT 1000
SAMPLE 1 a b 'Select an urn at random
IF b =1 'If urn selector says urn 1
SHUFFLE u1 u1 'shuffle urn 1
TAKE u1 1 c11 'take one coin from urn 1, call it c11
IF c11=6 'if the coin is a penny
TAKE u1 2 c12 'take the second coin
IF c12=6 'if the second coin is a penny
SCORE 6 z 'keep track of the second coin result
END 'end IF conditions
END
END
IF b =2 'if urn selector says urn 2
SHUFFLE u2 u2 'etc. as above
TAKE u2 1 c21
IF c21=6
TAKE u2 2 c22
IF c22=6
SCORE 6 z
END
END
END
IF b =3
SHUFFLE u3 u3
TAKE u3 1 c31
IF c31=6
TAKE u3 2 c32
IF c32=6
SCORE 6 z
END
END
END
END 'End the experiment, go back and
'repeat until 1000 repetitions
'are complete
COUNT z = 6 k
DIVIDE k 1000 kk
PRINT kk
Answer: KK = 0.34
Solution: Puzzle 2
1. Put a white ball (later have the computer call it "7" to
avoid confusion) or black (call it "8") in the urn with
probability .5.
2. Put in a white and shuffle the two balls.
3. Take out a ball. If black, stop and make no record.
4. (If result of (3) is white): Take out the remaining
ball, examine, and record its color.
5. Repeat steps 1-4 (say) until 1000 trials (with 1000
recordings) have been completed. (Alternatively, one can divide
the number of repetitions by the number of records in the
scoreboard).
6. Count the number and compute the proportion of whites
(7s) among the trials where the result of step (3) is white.
Carroll gives the answer as 2/3 (p. 32).
COPY (7 8) A '7=WHITE COUNTER, 8=BLACK COUNTER.
REPEAT 1000
SHUFFLE A B 'SHUFFLE THE TWO COUNTERS
TAKE B 1 C 'TAKE A COUNTER FOR THE BAG
CONCAT C 7 D 'JOIN A WHITE COUNTER TO THE BAG
SHUFFLE D E 'SHUFFLE THE BAG
TAKE E 1 F 'TAKE OUT A COUNTER
IF F =7 'IF THE COUNTER YOU TAKE OUT IS WHITE
TAKE E 2 G 'TAKE THE OTHER ONE
SCORE G Z 'RECORD THIS SECOND ONE'S COLOR
END
END 'END THE REPEAT LOOP, GO BACK AND REPEAT
COUNT Z =7 K 'COUNT HOW MANY TIMES SECOND ONE WAS WHITE
DIVIDE K ZZ KK 'EXPRESS AS A PROPORTION OF THE NUMBER OF
'TRIALS ON WHICH THE FIRST ONE WAS WHITE.
PRINT KK
Answer: KK = 0.66356
FROM PREVIOUS MESSAGE:
PROBLEMS:
Two Puzzles: Does your reasoning lead you astray on the following
puzzles? Most people's does. Here are resampling (simulation) solutions
that illustrate how such an approach, though less sophisticated than a
formulaic one, yields correct answers and offers fewer opportunities to go
wrong.
1) "Three identical boxes each contain two coins. In one
box both are pennies, in the second both are nickels,
and in the third there is one penny and one nickel.
A man chooses a box at random and takes out a coin. If
the coin is a penny, what is the probability that the
other coin in the box is also a penny?" [from Goldberg,
1960, p. 99]
2) A bag contains one counter, known to be either white or black. A white
counter is put in, the bag shaken, and a counter drawn out, which proves
to be white. What is now the chance of drawing a white counter? From
Lewis Carroll's PILLOW PROBLEMS (1895/1958) (p. 2, via Martin Gardner)