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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?

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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?

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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?

Odds and Evens

Stage: 3 and 4 Challenge Level: Challenge Level:1

You could start by using the interactivity - what happens when you pick out two balls from set $A$ $100$ times?
How many times did you get an even total? Were you expecting this?

Work out all the possible combinations for set $A$, to see if you can make sense of what happened.
How will you make sure you don't miss any combinations out?