In a game of
tictactoe, X
went first followed by O. The situation after two moves was as follows:
 
++
O  X 
++
 
Given that both X and O act at random, what is the probability that X wins the game?
(In reply to
computer aided solution by Charlie)
The following simulation verifies the probabilities found previously:
DECLARE SUB move (mno#)
DECLARE SUB moveww (mno#)
DECLARE FUNCTION fact# (x#)
DEFDBL AZ
CLEAR , , 25000
DIM SHARED grid(3, 3), wins(2)
grid(2, 2) = 1: grid(2, 1) = 2
CLS
FOR trial = 1 TO 100000
move 3
NEXT
FOR i = 0 TO 2
PRINT wins(i);
PRINT wins(i) / (wins(1) + wins(2) + wins(0))
NEXT
SUB move (mno)
player = (mno  1) MOD 2 + 1
psn = INT(9 * RND(1) + 1)
row = (psn  1) \ 3 + 1
col = (psn  1) MOD 3 + 1
WHILE grid(row, col) > 0
psn = INT(9 * RND(1) + 1)
row = (psn  1) \ 3 + 1
col = (psn  1) MOD 3 + 1
WEND
grid(row, col) = player
vhit = 1
FOR r = 1 TO 3
IF grid(r, col) <> player THEN vhit = 0: EXIT FOR
NEXT
hhit = 1
FOR c = 1 TO 3
IF grid(row, c) <> player THEN hhit = 0: EXIT FOR
NEXT
d1hit = 1
FOR r = 1 TO 3
IF grid(r, r) <> player THEN d1hit = 0: EXIT FOR
NEXT
d2hit = 1
FOR r = 1 TO 3
IF grid(4  r, r) <> player THEN d2hit = 0: EXIT FOR
NEXT
IF vhit OR hhit OR d1hit OR d2hit THEN
wins(player) = wins(player) + 1
ELSE
IF mno < 9 THEN
move mno + 1
ELSE
wins(0) = wins(0) + 1
END IF
END IF
grid(row, col) = 0
END SUB
The results:
11447 .11447 draw
72884 .72884 win for X
15669 .15669 win for Y

Posted by Charlie
on 20131224 21:45:24 