Batch 2 - Class 322 - Game of SKUNK (Probability)

• (in class)

• Rules of Skunk - Each letter of SKUNK represents a different round of the game; play begins with the "S" column and continue through the "K" column. The object of the game is to accumulate greatest points over five rounds. Rules:
• At beginning of each round, every player stands. Then, a pair of dice is rolled (common for all players)
• A player gets the total of the dice and records it in his or her column, unless a "6" comes up, in which case play is over for that round and all the player's points in that column are wiped out
• If "double 6" comes up, all points accumulated in prior column are also wiped out
• If "6" doesn't occur, the player may choose to continue standing and try for more points, or to stop and sit, in which case they get to keep what they have accumulated.
• If "6" or "double 6" comes on the first roll of a round, that round is over and all players bear the consequences
• Think about the turn and game strategy for this game.
• Try a practice game. At end, ask students what strategies they are trying. Try another practice game, and see how students are changing their strategy
• What is the chance and what is the choice in this game?
• How would you decide on whether to stand or sit in SKUNK?
• Since rolling a "1" is a disaster, it would be useful to know how many "good" rolls happen on an average before a "bad" roll. How can you find it out?
• Is this a good strategy or is there a better one?
• If a "1" isn't rolled, what is the average score you expect to get?
• Test it by rolling a pair of dice several times
• Ask students to describe the following strategies
• "Play it safe"
• Risky strategy
• What kind of scores might you get on both of above?
• Choose a strategy (some kids may do either of the two), play few games and analyze
• How well do the results agree with the objective (safe or risky)
• How does the strategy change if you are playing alone (to maximize average points per game) versus playing with competitors
• Let students create a 5-grade rating chart. How many points would they allocate for each grade range? Why?
• Suppose you are playing skunk but you are not told the outcomes of the dice each time. You have to decide when to sit based solely on the number of rolls so far. When should you sit?

• What are the similarities and differences in game of PIG versus game of SKUNK? Getting to 100 versus maximum score in 5 turns.
• Suppose that, before each roll, you are able to specify the number of dice that are to be rolled, and you can change this number from roll to roll based on your score. As before, the game is over for you, with zero payoff, if any of the dice show a 1. A strategy now must specify, for each score s, whether to remain standing and if so, how many dice to use. Find the optimal strategy.

• Should you break up your swim into multiple segments? Why or why not?
• Does it matter if you swim earlier or later?
• How do the above two answers help you simplify the problem?
• Does it ever make sense to partly swim and partly run? How do you think about that problem?
• Answer: the ratio of the length of an arc to that of the corresponding chord varies between pi/2, for a semicircle, down to a limit of 1 for tiny arcs. So if swimming an arc makes sense at all, it makes sense when the most distance is saved, that is, swimming directly across to the drink. You should do so if you can walk less than pi/2 times as fast as you swim.

• What if it is a rectangular pool - you are sitting at the center of one side, and the water is middle of opposite side. Specifically, can you think about any scenario where it might be better to swim and run.