# Each concert a singer makes profit of 0.75, but after every concert a singert can fall into bad mood

Each concert a singer makes profit of 0.75, but after every concert a singert can fall into bad mood with probability = 0.5. To get a singer out of these mood producer can send her flowers. If flowers cost $x$ money, then singer can get out of bad mood with probability $\sqrt{x}$. Which $x$ should producer choose in order to maximize his expected profit. (All the profit from concert goes to producer. One can make assumprion, that singer's career lasts $n$ days.)
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Yair Boyle
I'm not sure that this is the correct path, but one can solve it by brute force using backward induction.

Let $f\left(n\right)$ be the expected profit when there are $n$ days left and the singer is happy today and $g\left(n\right)$ be the expected profit when there are $n$ days left and the singer is sad today. I assume that if she's sad, she stays sad until flowers are sent and make her happy and there is one concert a day.

Clearly, $f\left(1\right)=0.75$ and $g\left(1\right)=\underset{0\le x\le 1}{max}\left\{\sqrt{x}0.75-x\right\}=\frac{9}{64}$.

For every $n>1$, $f\left(n\right)=0.75+\frac{f\left(n-1\right)+g\left(n-1\right)}{2}$ (no need for flowers, she's happy!) and $g\left(n\right)=\underset{0\le x\le 1}{max}\left\{\sqrt{x}\left[0.75+\frac{f\left(n-1\right)+g\left(n-1\right)}{2}\right]+\left(1-\sqrt{x}\right)g\left(n-1\right)-x\right\}$

I tried to calculate it for $n=2,3$ but it got ugly pretty quickly so I'm not sure there is a closed-form solution that can be guessed using this way.

Hope it helps!