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Snail Trails

This is a game for two players. You will need some small-square grid paper, a die and two felt-tip pens or highlighters. Players take turns to roll the die, then move that number of squares in a straight line. Move only vertically (up/down) or horizontally (across), never diagonally. You can cross over the other player's trails. You can trace over the top of the other player's trails. You can cross over a single trail of your own, but can never cross a pair of your trails (side-by-side) or trace over your own trail. To win, you must roll the exact number needed to finish in the target square. You can never pass through the target square. The game ends when a player ends his/her trail in the target square, OR when a player cannot move without breaking any of the rules.

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Game of PIG - Sixes

Can you beat Piggy in this simple dice game? Can you figure out Piggy's strategy, and is there a better one?

Who's the Winner?

Stage: 3 Challenge Level: Challenge Level:1

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When the teachers play the students at hockey, they are equally matched - at any point in the match, either team is equally likely to score. 
What are the possible results if 2 goals are scored in total?

Why are they not all equally likely?

This mathematical model assumes that when a goal is scored, the probabilities do
not change. Is this a reasonable assumption?
Alison suggests that after a team scores, they are then twice as likely to score the next goal as well, because they are feeling more confident. What are the probabilities of each result according to Alison's model?
Charlie thinks that after a team scores, the opposing team are twice as likely to score the next goal, because they start trying harder. What are the probabilities of each result according to Charlie's model?

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The models could apply to any team sport where a small number of goals are typically scored.

You could find some data for matches between closely matched teams that finished with two goals and see which model fits most closely to what happened.

You will need to make some assumptions about what it means for teams to be "closely matched".

Send us your conclusions, and explain the reasoning behind the assumptions you chose to make.