In the computer solitaire game "Freecell", statistics are kept for the largest winning streak, largest losing streak, and the percentage of games won.

Being mildly addicted to the game, I was not suprised to see that I had won 60 consecutive games while only losing 3 consecutive games.

Two questions based on this information:

1) What is my expected percentage of games won?

2) How many more games will I have to play before I have an even chance of a winning streak of 100 games?

There are a couple of problems trying to figure this out. We need to know the expected length of a run of successes or failures based on the probability of a win in each trial and the number of games played; and then, in order to get the expected value of the probability, given the lengths of the runs, we'd need to do a Bayesian analysis, going from the observed run lengths back to the probability of a win in a given game.

But if we seek only an approximation, we can look for the modal value of the individual probability of a win and number of games played that will lead to expected run lengths that match those observed. This is a modal value in that it is the one that gives the most likelihood of the observation, but it is not the average of the weighted probabilities that produce the observed outcome.

The following program simulates multiple trials, where ssize represents the number of games played and p is the probability of a win in an individual game. The values for these were varied until results matching the observations were found.

p = .9056
ssize = 2355
RANDOMIZE TIMER
FOR trial = 1 TO 500
  hct = 0: ctmax = 0: fct = 0: fctmax = 0
  FOR i = 1 TO ssize
    r = RND(1):
    IF r < p THEN
     hct = hct + 1: IF hct > ctmax THEN ctmax = hct
     fct = 0
    ELSE
     hct = 0
     fct = fct + 1: IF fct > fctmax THEN fctmax = fct
    END IF
  NEXT
  ctmTot = ctmTot + ctmax
  fctmTot = fctmTot + fctmax
  tct = tct + 1
NEXT
PRINT p; ssize; ctmTot / tct, fctmTot / tct, LOG(ssize) / LOG(1 / p)

Each row of the output represents 500 trials:

  p   ssize  avg max win  avg max lose   "predicted" max win streak
              streak        streak
.9056  2355  59.576         2.978         78.30261
.9056  2355  58.784         3.028         78.30261
.9056  2355  59.258         2.944         78.30261
.9056  2355  58.632         2.996         78.30261
.9056  2355  59.426         3.004         78.30261
.9056  2355  60.082         2.982         78.30261
.9056  2355  59.214         2.96          78.30261
.9056  2355  59.954         2.992         78.30261

The "predicted" max win streak column is based on a formula found on the web, searching for "expected longest run".  It clearly overestimates the average maximum winning streak.

So at this point, the best guess is that the probability of a win in a given game is about .9056, and that 2355 games had been played.

Leaving p the same, and increasing ssize we can seek how many games need to be played to expect a winning streak of 100 games:

ssize = 115000

results in an expected largest winning streak of about 100. But I've added a new column--what fraction of the time the largest streak is at least 100:

  p     games  exp max win exp max lose  fraction
                streak      streak     at least 100      web prediction
.9056  115000  99.578       4.622         .42           117.5168696365425
.9056  115000  99.994       4.616         .43           117.5168696365425
.9056  115000  99.728       4.614         .432          117.5168696365425
.9056  115000  99.694       4.614         .414          117.5168696365425
.9056  115000  100.212      4.626         .448          117.5168696365425

Note the web formula still overpredicts. More importantly, note that in only about 42% or 43% of cases does a streak of 100 appear. That's not "an even chance". This is an example of the mean not being the median.

If we increase the number of games in each trial to 139,000 or 140,000, we get about an equal possibility of achieving or not achieving a string or 100 wins, as given by the following reports:

.9056  139000  101.952      4.692         .498          119.4283908833486
.9056  139000  101.894      4.688         .498          119.4283908833486
.9056  139000  101.938      4.688         .488          119.4283908833486
.9056  139000  102.226      4.704         .502          119.4283908833486
.9056  139000  101.768      4.688         .492          119.4283908833486

.9056  140000  102.274      4.704         .504          119.500684802691
.9056  140000  101.962      4.688         .502          119.500684802691
.9056  140000  102.05       4.712         .494          119.500684802691
.9056  140000  102.108      4.694         .514          119.500684802691
.9056  140000  102.022      4.694         .506          119.500684802691

Again, each row represents the average of 500 trials of the given number of games. We see that the expected maximum winning streak is about 102, the mean.

The 139,000 or 140,000 begin now, as the first 2355, or whatever, played already,  have already been ascertained not to contain a 100-game winning streak.

Posted by Charlie on 2008-10-26 13:40:46