Consider the following game: at each round of game, $n^2$ cards, numbered from $1$ to $n^2$, are shuffled; you pick a card randomly. If you pick the card with number $1$ you win that round; each round costs one cent to play. Each time you win, you receive $\$100$. Would you play this game? Why or why not?
Would you play this card game?
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probability
probability-theory
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1Sorry, you only win if you pick card $1$? so you win with probability $\frac 1{n^2}$? – 2017-01-08
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0Assuming that I have understood the rules correctly...for a given $n$ compute the expected return, $E_n$. Here you win $99.99$ if you draw the $1$ and you lose $.01$ if you draw something else. I would argue that you should only play if your expectation is $>0$ but that's something of a subjective matter. – 2017-01-08
1 Answers
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Credit: thanks to lulu and Jorge for pointing out my mistakes.
Probability of winning, $\frac{1}{n^2}$, gain $\$99.99$
Probability of losing, $1-\frac1{n^2}$, damage $\$0.01$
Expected gain is $$\frac{99.99}{n^2}-\left(1-\frac{1}{n^2}\right)0.01=\frac{100}{n^2}-0.01$$
when will the expected gain be positive?
$$\frac{100}{n^2}>0.01$$ $$n^2<\frac{100}{0.01}=10000$$
$$n < 100$$
Hence if I were to just maximize my expected gain, I would play it if $n < 100$. However, it depends on how much risk you are willing to take. what is your utility function?
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1aren't the calculations slightly off? – 2017-01-08
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0yikes... thanks – 2017-01-08