A few of the popular board games where people makes inferences are Charades, Apples to Apples and HeadBandz. These games directly contain elements that require the player to use inferencing skills within the game.
A game that uses inference in a few different ways is Cranium Hoopla. Players are each dealt several cards at the beginning of the game. One player rolls a die that has several sides. Each side is a different color and depending on the top color the cube displays after the roll, the player will have to use some sort of inference to convey what picture is displayed on one of his cards. The other players then try to guess from the clues what the image on the card is. The player doing the activity has a choice of four things he can do to give the other players clues. He can draw pictures, act out a scene, use words that start with the same letter which hint at the answer, or create a simple riddle. The player must never use the words that specifically identify the picture, only clues to describe it. Whether the answer is guessed or not, play continues around, allowing all of the other players a turn at rolling the die.