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If I throw a 6-sided dice 50 times, what is the theoretical probability of getting fifty 2s?
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$P(X=50) = (\dfrac{1}{6})^{50} = 1.237 \times 10^{-39}$

Explanation:

The probability of getting a $2$ when a die is thrown is $\dfrac{1}{6}$

If we repeat the experiemnt many times, we can calculate the aggregate probability. We have to keep in mind the fact that each die-throw is an independent event and has nothing to do with the previous throw.

So if $\text{Pr}(2)=\dfrac{1}{6}$ then ${\text{Pr(not2)}}=\dfrac{5}{6}$

Using the binomial distribution

$P(A) = \binom{N}{k} \cdot p^kq^{N-k}$

$P(X=50) = \binom{50}{50} \cdot (\dfrac{1}{6})^{50}(\dfrac{1}{6})^{50-50}$

$P(X=50) = \binom{50}{50} \cdot (\dfrac{1}{6})^{50}$

$P(X=50) = (\dfrac{1}{6})^{50} = 1.237 \times 10^{-39}$

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