A game is played starting with 6 fair coins laid out with heads face up.
Each round consists of flipping all of the coins showing heads.
If an even number of coins is flipped and an even number of heads comes up the player loses.
If an odd number of coins is flipped and an odd number of heads comes up the player loses.
Rounds continue until the player either loses or reaches zero heads.
The player wins by getting to zero heads without losing.
What is the probability of winning this game?
Examples:
6→3→1 would be a loss.
6→3→0 would be a win.
6→0 would be a loss (even→even rule is applied first.)
5 point 10
9 for Nbr=6 to 1 step 1
10 print Nbr,1fnPLose(Nbr),(1fnPLose(Nbr))/1
11 next Nbr
20 end
30
40 fnPLose(N)
50 local Pl,H,Ph,Totph
60 Pl=0:if N=0 then goto 150
70 for H=0 to N
80 Ph=combi(N,H)//2^N
90 if H @ 2<>N @ 2 then
100 :Pl=Pl+Ph*fnPLose(H)
110 :else
120 :Pl=Pl+Ph
130 :endif
140 next H
150 return(Pl)
The above program, working recursively, again fills in Steve Herman's list of probabilites when starting with different numbers of tossed coins. In descending order, they are:
6 16793/131072 0.12812042236328125
5 563/4096 0.137451171875
4 23/128 0.1796875
3 7/32 0.21875
2 1/4 0.25
1 1/2 0.5
and, since the puzzle starts with 6 coins, the solution is 16793/131072 = 0.12812042236328125. All the decimal fractions are exact and terminating as all the denominators are powers of 2.
Simulation verification:
A million trials, via
FOR tr = 1 TO 1000000
n = 6
loss = 0: win = 0
DO
hds = 0
FOR c = 1 TO n
h = INT(RND(1) * 2)
hds = hds + h
NEXT
IF hds MOD 2 = n MOD 2 THEN loss = 1 ELSE IF hds = 0 THEN win = 1
n = hds
LOOP UNTIL loss = 1 OR win = 1
losses = losses + loss
wins = wins + win
PRINT wins, losses, tr
NEXT tr
result in 127731 wins.

Posted by Charlie
on 20110128 18:15:08 