Everyone knows Candyland, right? It's a kids game with no choices. You just keep spinning or drawing cards and you do what they say and it takes a long time and it's hard to say how much longer it is going to take. See more comments along these lines in my analysis of Hi Ho! Cherry-O.
There are a bunch of versions, not even to mention the spin-offs. We have a recent edition (as of 2017) that uses cards. Other versions use a spinner. In our version, there are pitfalls that require you to draw a particular color card to proceed. I see that in other versions, they simply make you lose one turn. Here's an analysis of that version, which finds that the games are much shorter that way (but I haven't checked if that's the only difference in the analysis). I don't know how many mechanically distinct versions there are.
In my set, the deck is:
It's not clear that having 8 of some of the single colors and 7 of others is intentional. The above totals to 63 cards, but the rules say there are 64 cards. What would the last one be? A previous Monte Carlo analysis I found on the web assumed there were 66 cards, with 8 of each single color, which actually makes 67. So different versions may have different decks and/or quality control isn't very good. Maybe they figure kids are going to lose or destroy cards anyway…
Here's some Monte Carlo code. Let's see how many turns it takes to finish a game and who wins:
| Players | #1 wins | #2 wins | #3 wins | #4 wins | Mean plies | Median plies |
|---|---|---|---|---|---|---|
| 1 | 100% | — | — | — | 44 | 37 |
| 2 | 50.6% | 49.4% | — | — | 55 | 49 |
| 3 | 34.0% | 33.3% | 32.7% | — | 65 | 59 |
| 4 | 25.7% | 25.3% | 24.8% | 24.2% | 75 | 68 |
where a "ply" is one turn taken by one player.
There are some details left out of the printed rules that must be implemented. One is how to handle the deck running out. I chose the simplest option, which is that you run through the entire deck and then shuffle. In actual play, one might pre-emptively shuffle the discard pile, but not here. This mostly affects the distribution of long jump cards coming up.
Some people find it unclear how to get to the winning space. I've implemented what I think is the usual interpretation, which is that if you draw a color and there are no spaces ahead of you with that color, you go to the winning space and win.
Anyway, back to the analysis. Unlike Hi Ho Cherry-O, this game has a deck, and therefore has a fair amount of memory of what has happened so far. Notably, there are cards that warp you to one of seven places on the board, but once you've drawn one of these, you can't get it again until the deck is shuffled. Early on, these tend to move you forward, but later on the reverse is true. Also of minor note is the fact that you can't be reset all the way to the first square, so the very beginning of the game is special.
Here's the distribution for a 1-player game:
1.92% | _--_
1.84% | --_ _~ ~-_
1.76% | ~ ~~~ -_
1.68% | - ~_
1.60% | ~-_
Prob 1.52% | - ~_
of 1.44% | _ ~-_
End 1.36% | -_
1.28% | - ~-_
1.20% | ~_
1.12% | ~ ~-_
1.04% | ~-_
0.96% | - ~-_
0.88% | ~-_
0.80% | ~-_
0.72% | - ~_
0.64% | ~_
0.56% | ~-
0.48% | ~-_
0.40% | ~ ~--_
0.32% | ~~--___
0.24% | ~~~~~-----_____
0.16% | ~ ~~~~~~------_______
0.08% | _ ~~~~~~~~-----_________
0.00% |----- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-------------------
++---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190
Turns
And to more clearly show the periodic structure, this is the probability of finishing on a given turn given that you haven't finished yet:
4.50% | _
4.20% | __--~~~-_ ___~-~ ~-_
3.90% | __---~~--__ ___---~~ ~-_ ___--~ ~-_
3.60% | __---~~~ ~- ____--~~ ~-_ ____-~~~~ --
3.30% | ___---~~~~ ~--_ ___---~~~ ~-__ __----~~~ ~~
Prob 3.00% | ____---~~~ ~~~---_______----~~~~ ~~---______---~~~~~
of 2.70% | _-~~~
End 2.40% | _-~
2.10% | _----~~
1.80% | -
1.50% | -~
1.20% | -~
0.90% | ~
0.60% | ~
0.30% | _~
0.00% |------
++---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190
Turns
The peak at 16 turns is real, not a statistical artifact. It is formed out of games in which no backwards jumps happen. Here's the distribution of just those such games:
1.38% | _-
1.32% | _ ~
1.26% | _
1.20% | -~
1.14% | -
1.08% | -
1.02% | -
0.96% | _
Prob 0.90% | ~
of 0.84% | ~_
End 0.78% | -
0.72% | - ~-
0.66% | _
0.60% |
0.54% | ~
0.48% | -
0.42% | ~ _
0.36% |
0.30% | ~_
0.24% | -
0.18% | ~ ~-
0.12% | ~--_
0.06% | _ ~~--____
0.00% |----- ~~~~~~----------
++---------+---------+---------+---------+---------+---------+---
0 10 20 30 40 50 60
Turns
This allows pitfalls (where you get stuck and can't move). If those are also disallowed, the shoulder around 25 mostly goes away.
The shortest possible 1-player game is 4 turns, for instance {ice cream, double orange, double blue, any double}. However, it's not very likely. To begin with, the first draw must be ice cream, which is a 1/63 chance. The probability is 3×10−5.
While the rules say that it is a game for 2–4 players, there is clearly nothing stopping more people from playing. The mean number of turns per player decreases steadily as you add players, since it only takes one lucky player to end the game. Once you reach about 500 players, it becomes very likely that someone manages a 6-turn win, via the ice cream followed by many combinations of four colors and a double-color, e.g. {ice cream, double yellow, purple, green, green, orange}, or better.