Solutions - 5070-07

(a)

(1) Count total outcomes:

Rolling a die $100$ times gives $6^{100}$ total outcomes.

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(2) Count desired outcomes:

We choose 16 of the 100 to be ones, 17 of the remaining 84 to be twos, and the other $67$ can be any of $3,4,5,6$, giving $4^{67}$ arrangements.

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(3) Compute probability:

$$\frac{\left(\genfrac{}{}{0px}{}{100}{16}\right)\left(\genfrac{}{}{0px}{}{84}{17}\right)4^{67}}{6^{100}}$$

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(b)

(1) Count desired outcomes:

We choose 25 of the 100 to be ones. Out of the remaining 75 rolls, we choose at least 50 to be twos. The remaining $75-k$ rolls can be any of $3,4,5,6$.

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(2) Compute probability:

$$\sum _{k=50}^{75}\frac{\left(\genfrac{}{}{0px}{}{100}{25}\right)\left(\genfrac{}{}{0px}{}{75}{k}\right)4^{75-k}}{6^{100}}$$