If 30 ice-cream sticks are numbered 1-30 and placed in a bowl, what are the odds that when picking two at a time, and then returning them to the bowl, any random pair will never be picked?
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The total number of pairs is $30 \times 29 / 2 = 435$ (30 for the first member of the pair, 29 for the second, and divided by 2 because order doesn't matter). So the probability of drawing a specific pair in a single draw is $1/435 \approx .002$. The probability of not drawing a specific pair is $434/435$, so the probability of not drawing a specific pair in $k$ independent draws with replacement is $(434/435)^k$. Odds are just $p/(1-p)$ (where $p$ is the probability), so the odds of not drawing the pair in $k$ independent draws with replacement is $\frac{(434/435)^k}{1-(434/435)^k}$.
Noah
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