Odds of Success

When considering game design, one should think about how often the avatar/PC/player should succeed. If the players fail too often, then they will get frustrated. If they succeed all the time, they will get bored. Finally, there's a connection to reality. Depending on what the game is trying to model, wildly bizarre results may strain credulity; for example, it may not be fun to play a game that gives an untrained but very dexterous individual a 95% chance of success for, say, brain surgery.

Casinos are well aware of this. For example:
"Customers at slot machines receive frequent, small prizes at irregular intervals. This encourages continued betting in the hope of getting a big jackpot." (http://www.scientificpsychic.com/workbook/gambling.html). Scientists have looked into this phenomenon to examine the frequency of wins that a player needs to receive to stay "hooked." Here is one such paper (http://www.camh.net/egambling/archive/pdf/JGI-issue11/JGI-Issue11-turner-horbay.pdf), and another (http://www.camh.net/egambling/issue21/04haw.htm).

It looks like many slot machines reinforce about 1/6 to 1/3 pulls (that is, you win 1/3 to 1/6 of the time, at a minimum). In blackjack, I found these odds: Odds of dealer winning - 45.85% Odds of a tie - 9.53% Odds of a player winning - 44.62%. Additionally, consider the role of "near misses" -- that, is times when the player ALMOST wins. One more factor is that of nearby wins -- notice that slot machines are always in a row of at least 5? That guarantees that with 3 players sitting there (man space...), one will be winning almost every pull. At a blackjack table with 4 players, someone will win almost every hand (obviously sometimes the dealer gets a 21 or busts out everyone, but that doesn't happen every time).

Switching this to gaming, a Dragonsfoot Poll asked the question, "% chance of success when attempting a task should be:" is to gaming (http://www.dragonsfoot.org/forums/viewtopic.php?f=11&t=27372&hilit=chance+to+succeed)

The results are reproduced here:
33% trained / 0% untrained (i.e.: d6 roll, 1-2 tr but no untr) 0% [ 0 ]
50% trained / 0% untrained 7% [ 2 ]
45% trained / 15% untrained (i.e.: base Siege Engine check) 10% [ 3 ]
66% trained / 33% untrained (i.e.: d6 roll, 1-4 tr or 1-2 untr) 41% [ 12]
75% trained / 25% untrained 17% [ 5 ]
other 24% [ 7 ]

Note that the most popular result is 66/33. Remember our slot machine odds? Pretty close (for the untrained figure, at least!).

Now let's look at some real world figures. The Armed Forces have requirements for marskmanship with various weapons. These requirements have two objectives that I can think of: (1) practically, to ensure that soldiers can fire their primary weapon with some degree of success and (2) psychologically, to encourage the soldier to feel confident in his ability to use lethal force, and provide a means to reward him for being accurate and eager with his weapon. The services don't want to set the bar too high, or else nobody will pass, which is psychologically damaging and bad for morale -- plus its expensive to have all those remedial fires! You can't set the bar too low, though, or else it poses a danger to the improperly trained soldiers themselves and makes those who are actually trained frustrated because its too easy to succeed.

Here's the AF standards:

EXPERT: 43/50 0.86
A: 32/50 0.64
B: 25/50 0.5
C: 19/50 0.38

Note that group A are people who need to use the skill on a regular basis (security forces, for example), so they have a higher standard. Arming group C are people who do not need to shoot often (office employees), so they have a lower standard. Expert is a meritorious category that anyone can be rewarded for attaining. Note the Category A figures falling right around that magic 2/3 area identified in our DF poll which is good for a "trained" person. Category C falls just about dead on our 1/3 area used for "untrained" persons, and also matches up with our untrained slot machine chances of winning.

Here's the Army Standards:
EXPERT: 36/40 0.9
SHARPSHOOTER: 30/40 0.75
QUAL: 23/40 0.575

In the Army, everyone uses the same standard, but there are two tiers of meritorious performance instead of just one like the USAF "Expert" category. Note the qual and sharpshooter figures falling right around that magic 2/3 area identified in our DF poll, again good for a "trained" person.

Obviously one would need to do a study to find the ideal frequency of wins for addiction/satisfaction, but I think the 66/33 rule of thumb is a good rough hack. It ensures that in a group of ~5 players, one will succeed at the task most of the time (if all are untrained, at least one will succeed about 87% of the time). It matches up with real world marksmanship odds. Its easy to generate on standard dice. It rewards training without making untrained attempts absolutely fruitless.