Respuesta :
If you start with the same number of red and black cards, you win with probability 1/2 hence you should pay nothing to play.
Explain same number of red and black cards, you win with probability 1/2 hence you should pay nothing to play?
If there are r red cards on a total of n cards let p(r,n) be the probability to win with the optimal strategy. If you stop immediately you win with probability r/n. If you don't stop you can get a red card with probability r/n and then you win with probability p(r−1,n−1). If instead you get a black card you win with probability p(r,n−1). Hence with the best strategy you win with probability
p(r,n) = max {[tex]\frac{r}{n}[/tex], [tex]\frac{r}{n}[/tex] p(r-1), n-1) +[tex]\frac{n-r}{n}[/tex] p( r,n-1 }
Let's prove by induction that p(r,n) = [tex]\frac{r}{n}[/tex]. Suppose it is true for n−1 that p(r,n−1) = r/(n−1) then
p(r,n)=max {[tex]\frac{r}{n}[/tex], [tex]\frac{r}{n}[/tex]} [tex]\frac{r-1}{n-1}[/tex] +[tex]\frac{n-r}{n}[/tex] p (r,n-1)}
In the case n=1 you immediately find that p(0,1)=0 and p(1,1)=1.
So at anytime you stop you always get the same probability to win.
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