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Coin Tossing Games

You and I play a game involving successive throws of a fair coin. Suppose I pick HH and you pick TH. The coin is thrown repeatedly until we see either two heads in a row (I win) or a tail followed by a head (you win). What is the probability that you win?

Win or Lose?

A gambler bets half the money in his pocket on the toss of a coin, winning an equal amount for a head and losing his money if the result is a tail. After 2n plays he has won exactly n times. Has he more money than he started with?

Thank Your Lucky Stars

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand corner of the grid?

Who's the Winner?

Age 14 to 16
Challenge Level

hockey sticks
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?

hockey sticks

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.