You look at a carnival game. The person running it says, "Just reach your hand into this bag. There are 9 yellow balls and 1 red ball in the bag. You get 4 chances to pull out the red ball. (You have to put the ball you drew back before you draw another ball.) You only have to pay one dollar to play, and you get 3 dollars if you pull out the red ball!"

Assuming the person running the game is telling the truth, and the balls only differ in color, would you expect to make a net profit or a net loss on this game?

(In reply to re(4): Solution by Brian Wainscott)

Backing up Bryan's comments on needing to be able to ride out losing streaks, a simulation shows that, for example, if the player starts with a bankroll of $10 and a goal of playing until that has grown to $100, in 100,000 simulations of this situation, the player got to the $100 only 28% of the time, having lost the whole $10 72% of the time. It took an average of about 1400 plays to win when she did so, and 243 plays to lose when that happened.

When greedier, and going for $1000, she still lost about the same percentage of the time, but took on average about 343 plays to lose, and about 30,000 plays to win.

And when starting out with a measly $1, trying to leave with $100, she won not much more than 3% of the time (but of course gaining $100) and took an average of about 24 plays for a loss or 1400 plays for a win.

So, yes, losing streaks are statistically "inevitable" (or rather, merely likely), and the house has the "advantage" of a bigger bankroll against an individual, though with these odds not against the public at large.

The results (first line in each pair showing initial bankroll and goal; second line showing the number of trials, number of wins, number of losses, average number of plays for a win, and average number of plays for a loss) of several trials are:

 10            100


 100000        28133         71867         1419.564      242.7117


 10            1000


 1000          275           725           29941.54      343.24


 1             100


 100000        3288          96712         1425.148      24.40842

-----------
RANDOMIZE TIMER
res0 = 1
goal = 100
FOR trial = 1 TO 100000
 res1 = res0
 tot = 0
 DO
   IF RND(1) < .3439 THEN
    res1 = res1 + 2
   ELSE
    res1 = res1 - 1
   END IF
   tot = tot + 1
 LOOP UNTIL res1 < 1 OR res1 >= goal
 IF res1 < 1 THEN
   loss = loss + 1:
   ltot = ltot + tot
  ELSE
   win = win + 1
   wtot = wtot + tot
 END IF
NEXT trial
PRINT res0, goal
PRINT trial - 1, win, loss, wtot / win, ltot / loss

Posted by Charlie on 2003-09-18 17:25:04