Person A and Person B toss a fair coin. Person A wins a single toss when the coin falls on heads, and person B wins a single toss when the coin falls on tails. The goal of A is to win twice (total in the game), and the goal of B is to win 3 times. The grand winner is the person who reaches their goal first.
- What is the probability that A wins? (P(A))?
I already calculated this one by drawing a probability tree and reaching the answer of $11/16$ the tree
- What is the estimated value in the game, E(X)?
I'm not sure if I should do an average in the tree of the max tosses of 4 or not, please help
- Lets mark with P(n,m) the probability that A wins the game when his goal is n and B's goal is m. Using ${\displaystyle \ P(B)=\sum _{i}P(B|A_{i})P(A_{i})}{\displaystyle \ P(B)=\sum _{i}P(B|A_{i})P(A_{i})}$, find $\ F_{{n+1}}=F_{n}+F_{{n-1}}$ for $P(n,m)$.
I could use a direction on how to start.
To get the expected value you need to define the payoff. If $A$ wins $1$ on winning and loses $1$ on losing his expected value is $\frac {11}{16}(+1)+\frac 5{16}(-1)=\frac 6{16}$