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Let $X$ be a discrete random variable with probability function $P(X=x) = \frac{2}{3^x}$ for $x = 1,2,3,\ldots$

What is the probability that $X$ is even?

I have: $$\frac2{3^2}+\frac2{3^4}+\frac2{3^6}+\ldots$$ which is a geometric series of the form $$\sum_{n=1}^\infty ar^n$$ where $a = \frac29$ and $r=\frac19$.

Then I used the formula for finding the sum of a geometric series to find the sum/probability \begin{align} \sum_{n=1}^\infty ar^n &= \frac{a}{1-r} \\ &= \frac{\frac29}{1-\frac19} \\ &= \frac{2/9}{8/9} = \frac28 = \frac14 \end{align}

Correct?

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Yes, the summation is correct. –  André Nicolas Oct 25 '13 at 20:26
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Don't deface your questions. –  Potato Oct 25 '13 at 20:39

2 Answers 2

Following @Henry, your approach is just fine. For variety, here is yet another. In trinary, you have $$0.020202020\ldots$$ Is it easy to see that three ($10$ in trinary) times this is $$0.20202020\ldots$$ and then summing the two gives $$0.222222222\ldots$$ which is $1$? Therefore four times your number is $1$.

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Your method is correct and is a reasonable approach.

Here is another:

  • you have here $\Pr(X=2n) = \frac13 \Pr(X=2n-1)$ for all integer $n$,

  • so the probability of $X$ being even is a third of the probability of $X$ being not even,

  • making the probability of $X$ being even equal to $\frac14$.

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