LCR - Probability
LCR is not a game where the player makes choices. He or she can only either decide to play or not to play. Thus knowing the odds is useful (unlike calculating probability in poker) only so far as it gives you an idea of what is going on during play.
For a player to come up with three of L, C, or R is fairly unusual. Only one 1 in 72 rolls will result in any particular one of these combinations (three L's, three C's, or three R's). We can figure this out by knowing that there are 3 dice and the chances of getting any one of the three on the first of them is 3/6 (reduced to 1/2). On the second die the chances are also 1/6, and so on. To find our chances then, we multiply all three together:
1/2 X 1/6 X 1/6 = 1/72
The chances of retaining (three dots) or losing (any combination of L,C, and R) all of your chips on a roll of three dice is determined in a similar way:
3/6 X 3/6 X 3/6 = 27/216 or 1/8
The chances of losing 3 chips all at once are exactly the same as retaining all three because there are three ways to lose your chips and three spots on the dice to allow you to retain them. Once you are out of chips, your chances of coming back into chips are fairly good. If the player to your right or your left has 1 chip the chances are 1 in 6. If the player has two chips, your chance is 11 in 36 (getting better). And if the player has 3 dice, the chances are
91 in 216 (even better yet).
The way to figure out this last item is a bit complicated, but we'll take you through the process. What we really want to know is what are the chances of getting at least one L out of three independent dice rolled together. To do this we must find out what the chances of getting three L's out of three dice (we did this above, 1/216). Then two out of three. This is a bit trickier:
(1/6 X 1/6 X 5/6) X 3 = 15/216
The 5/6 is the possibility of getting anything but an L on one of the dice. The thing to remember here is that we are looking for the number of ways two L's might turn up on a roll of three dice. The chances of any particular permutation are exactly even. Thus we can take the number of ways and divide it by the number of total possibilites to get our chances of there being 2 L's. We multiply by three to take into account the fact that the 5/6 could be in the first second or third position, giving us 3 times the permutations. We do a very similar calculation to find out what the chances of getting only one L.
(1/6 X 5/6 X 5/6) X 3 = 75/216
By adding these three numbers together, 1/216 + 15/216 + 75/216 = 91/216, we come up with our result.
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