Show that the probability of scroing exactly n points is

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Homework Statement


A player tosses a coin repeatedly. Heads is one point, tails is two points. A player tosses until his score equals or exceeds n. Show that the probability of scoring exactly n points is (2+(-1/2)^n)/3


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The Attempt at a Solution


My guess would be a proof by induction, but not really sure how to go about this or any attempted proof

Thanks
 
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Seems pretty straightforward to me. Why don't you start by doing the initial cases (n=1,2) and writing out the inductive step?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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