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I have a bag with balls numbered 1 to 100. I take out a ball and put it back in the bag. I repeat the experiment 14 times. What is the probability that I get the same number twice?

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Answer to a math question I have a bag with balls numbered 1 to 100. I take out a ball and put it back in the bag. I repeat the experiment 14 times. What is the probability that I get the same number twice?

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Tiffany
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Para encontrar la probabilidad de que te salga dos veces el mismo número, debemos considerar primero el total de posibles resultados y después contar cuántos de esos resultados cumplen con la condición requerida.

Para cada una de las 14 extracciones, hay 100 números posibles que pueden salir. Por lo tanto, el total de posibles resultados distintos es de $100^{14}$.

Para contar cuántos de esos resultados cumplen con la condición requerida, primero elegimos un número que se repetirá dos veces, hay 100 maneras de hacerlo. Para las dos extracciones del mismo número, hay $1$ manera para la primera y $1$ manera para la segunda. Para los otros 12 números, hay $99^{12}$ maneras de que aparezcan en las 12 extracciones restantes.

Entonces, el número total de resultados que cumplen con la condición requerida es $100 \cdot 1 \cdot 1 \cdot 99^{12}$.

La probabilidad de que te salga dos veces el mismo número es:

\frac{100 \cdot 1 \cdot 1 \cdot 99^{12}}{100^{14}} = \frac{99^{12}}{100^{13}} \approx 0.37\%

Entonces, la probabilidad de que te salga dos veces el mismo número en las 14 extracciones es aproximadamente $0.37\%$.

$\boxed{0.37\%}$

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