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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?

Last One Standing

Stage: 4 Challenge Level: Challenge Level:1

Imagine a school assembly with 250 students. Everyone stands up and flips a coin. People with tails sit down. People with heads flip again.
Do you think anyone will get 6 heads in a row?
How many heads in a row do you expect the last one standing to have flipped?
Can you explain your reasoning?
Here is an animation for you to explore what happens when different sizes of assembly gather and carry out the experiment.

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Now that  you have had the chance to explore, do your answers and reasoning to the questions above change at all?
What is the probability of flipping ten heads in a row?
How many people would you need to have in a school assembly for you to expect there to be someone still standing after ten flips?
Here are some related questions you might like to consider:
The probability of winning the lottery jackpot if you buy one ticket is approximately 1 in 14 million. 
There are usually two jackpot winners every week. How many tickets do you think are sold each week? 

On October 7th 2010, a woman gave birth to her third child. Her first two children were also born on October 7th, in 2005 and 2007. So all three children in the family have the same birthday. The odds of this happening were incorrectly reported in the newspapers as being 1 in 48 million. Can you work out the correct probability?
There are more than a million families in the UK with three children.
Would you expect there to be other families with three children who share a birthday?
The television performer Derren Brown once filmed himself flipping ten heads in a row for a programme about horse racing and unlikely events. He used a fair coin, and kept filming until he got ten in a row. How long do you think it took him?