Here’s a good one on long-term probability and “beating the house”.1
Pictured above are “pig dice”, which land in six different positions. One of the positions features the pig landing on all four (we’ll simply refer to this as “feet”).
Here is the scenario:
In a game of chance, you will choose a number of times to roll a pig die. If you manage to roll “feet” on at least 25% of your rolls, you win a prize!2
Here are your choices:
I. 4 rolls
II. 24 rolls
III. 64 rollsWhich of the 3 preceding choices gives you the BEST chance of winning a prize? (In other words, in which scenario are you MOST likely to roll at least 25% “feet”?)
(At this point, I ask students to pick one… then take a show of hands. Results are usually mixed.)
I happen to have a whole bagful of these little pig dice, so I’ll let each student grab one, and start taking some rolls. Have the students track their proportion of “feet” after each roll, both in a table and on a graph, as so:
We will usually stop at 4 rolls, at which point I will ask students to raise their hand if they are a “winner” (aka, have at least 25% “feet”… which only requires one “feet” at this point). Typically at least half of the class is a winner at this early juncture.
Then we continue rolling…
By the time we stop again at 24 rolls, the number of “winners” has dwindled down to about 3 or 4 students. And of course, that’s the point.
We’re lucky to have even one winner by the time we get to fifty rolls.
If you ask the students to tell you what percentage they seem to be approaching, it’s usually somewhere between 0.05 and 0.15. Empirical data3 suggests that these pig dice land on feet around 10% of the time, give-or-take.
Thus is the Law of Large Numbers: long-term probabilities will reveal true behavior if you let time run its course. 4
Moral of the story: If you want to beat the house, win quick and GET OUT.