Weighted coin toss
I have been given a two pronged problem to solve at college whereby I need to calculate the percentages involved in winning a game with a weighted coin.
The game is the be the first to score 5 successes and the coin is weighted 70/30 in favour of heads.
First part is:
How would I go about calculating the possible scorelines involved and the percentage chances of either side winning this game?
I have tried using binomial distribution by drawing out a map and checking off potential scorelines but get confused when having to nullify branches of the tree that are essentially out of play (becuase the goal of 5 has already been reached).
Is there a more simple approach to working out percentages for all possible scorelines and the chances of winning the games?
The Second and more tricky part is:
How do I calculate the percentage chance of either side winning in any given scoreline?
So for example if it is 3-1 to Heads or they are drawing 4-4 , what are the chances of either side winning now?
Any help would be appreciated
How is the game played? What is a success? Do you mean that one person is tails and the other person is heads and whoever gets 5 of their side of the coin first is the winner of the game?
Originally Posted by Ken Holloway
Please post the question exactly as it's written because what you've posted is too vague (for me, anyway).
Hi Mr Fantastic, thanks for responding.
A success is simply tossing a head or a tail. So if the first three tosses are HEAD , TAIL ,HEAD then the current score is 2-1 to Heads. In a best of 9 game (or first to score 5), heads would only need 3 more successes to win the game while tails would need 4.
The total scorelines available in this example would be:
5-0, 5-1, 5-2, 5-3, 5-4 and 0-5, 1-5, 2-5, 3-5, 4-5
I'm trying to find a simple way of calculating the percentage chance of a head or tail winning a game and all of the possible scorelines (I imagine that if I can calculate any given scoreline then I can add them up and that will give me the chances of winning the game for each side).
On each game the weighting of the coin can be different and so can the goal (first to 5, 7 ,9 whatever)
The second and more tricky part is to be able to calculate the percentage chance of either side winning at any stage of the game. So if it is 3-1 to Heads what does that do either sides overall percentage chance of winning the game?
I have tried using binomial distribution but have gotten stuck on the fact that certain branches of the tree are no longer active (like when it hits 5-0 for example). Any help would be appreciated.