Question

# You have a six-sided die that you roll once. Let $R_{i}$ denote the event that the roll is i. Let $G_{j}$ denote the event that the roll is greater than j. Let E denote the event that the roll of the die is even-numbered. (a) What is $\mathrm{P}\left[R_{3} | G_{1}\right]$, the conditional probability that 3 is rolled given that the roll is greater than 1? (b) What is the conditional probability that 6 is rolled given that the roll is greater than 3? (c) What is $\mathrm{P}\left[G_{3} | E\right]$, the conditional probability that the roll is greater than 3 given that the roll is even? (d) Given that the roll is greater than 3, what is the conditional probability that the roll is even?

Solution

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Sample Space for rolling a die, S = {1, 2, 3, 4, 5, 6}

Since, all the outcomes are equally likely, P(rolling any number) = $\dfrac{1}{6}$

$\textbf{(a)}$

As we can see, $G_{1}$ is the sample space in this case.

$G_{1}$ = {2, 3, 4, 5, 6}

Hence, P($G_{1}$) = P($R_{2}$) + P($R_{3}$) + P($R_{4}$) + P($R_{5}$) + P($R_{6}$)

Hence, P($G_{1}$) = $\dfrac{1}{6} + \dfrac{1}{6} + \dfrac{1}{6} + \dfrac{1}{6} + \dfrac{1}{6} = \dfrac{5}{6}$

and $R_{3}$ = {3}

Hence, $G_{1} \cap R_{3}$ = {3} = $R_{3}$

Hence, P($G_{1} \cap R_{3}$) = P($R_{3}$) = $\dfrac{1}{6}$

Hence, the conditional probability that 3 is rolled given that the roll is greater than 1 i.e.

P($R_{3} | G_{1}$) = $\dfrac{P(G_{1} \cap R_{3})}{P(G_{1})}$ = $\dfrac{\frac{1}{6}}{\frac{5}{6}}$ = $\boxed{0.2}$

$\textbf{(b)}$

As we can see, $G_{3}$ is the sample space in this case.

$G_{3}$ = {4, 5, 6}

Hence, P($G_{3}$) = P($R_{4}$) + P($R_{5}$) + P($R_{6}$)

Hence, P($G_{3}$) = $\dfrac{1}{6} + \dfrac{1}{6} + \dfrac{1}{6} = \dfrac{3}{6}$

and $R_{6}$ = {6}

Hence, $G_{3} \cap R_{6}$ = {6} = $R_{6}$

Hence, P($G_{3} \cap R_{6}$) = P($R_{6}$) = $\dfrac{1}{3}$

Hence, the conditional probability that 6 is rolled given that the roll is greater than 3 i.e.

P($R_{6} | G_{3}$) = $\dfrac{P(G_{3} \cap R_{6})}{P(G_{3})}$ = $\dfrac{\frac{1}{6}}{\frac{3}{6}}$ = $\boxed{0.33}$

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