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Two dice are rolled. Let X = 1 if the dice come up doubles
Chapter 4, Problem 6E(choose chapter or problem)
Two dice are rolled. Let \(X = 1\) if the dice come up doubles and let \(X = 0\) otherwise. Let \(Y = 1\) if the sum is 6, and let \(Y = 0\) otherwise. Let \(Z = 1\) if the dice come up both doubles and with a sum of 6 (that is, double 3), and let \(Z = 0\) otherwise.
a. Let \(p_{X}\) denote the success probability for \(X\). Find \(p_{X}\) .
b. Let \(p_{Y}\) denote the success probability for \(Y\) . Find \(p_{Y}\).
c. Let \(p_{Z}\) denote the success probability for \(Z\). Find \(p_{Z}\).
d. Are \(X\) and \(Y\) independent?
e. Does \(p_{Z}=p_{X} p_{Y}\)?
f. Does \(Z = XY\)? Explain.
Questions & Answers
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QUESTION:
Two dice are rolled. Let \(X = 1\) if the dice come up doubles and let \(X = 0\) otherwise. Let \(Y = 1\) if the sum is 6, and let \(Y = 0\) otherwise. Let \(Z = 1\) if the dice come up both doubles and with a sum of 6 (that is, double 3), and let \(Z = 0\) otherwise.
a. Let \(p_{X}\) denote the success probability for \(X\). Find \(p_{X}\) .
b. Let \(p_{Y}\) denote the success probability for \(Y\) . Find \(p_{Y}\).
c. Let \(p_{Z}\) denote the success probability for \(Z\). Find \(p_{Z}\).
d. Are \(X\) and \(Y\) independent?
e. Does \(p_{Z}=p_{X} p_{Y}\)?
f. Does \(Z = XY\)? Explain.
ANSWER:Step 1 of 6
a)
Let \(X = \left\{ \begin{array}{l}{\rm{1}}\left( {{\rm{success}}} \right),\;\;\;{\rm{dice}}\;{\rm{comes}}\;{\rm{up}}\;{\rm{doubles}}\\0\left( {{\rm{failure}}} \right),\;\;\;\;\;{\rm{Otherwise}}\end{array} \right.\)
The outcomes of rolling two dice are pairs \(\left( {{n_1},{n_2}} \right)\) where \({n_1}\) is the number from 1 to 6 the first die comes up and \({n_2}\) is the number from 1 to 6 the second die comes up.
There are (6) (6) = 36 of such pairs. Since X = 1 when the dice come double, the outcomes that correspond to the success are (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6). Therefore, the success probability is,
\({p_x} = P\left( {X = 1} \right)\)
\({p_x} = 6/36\)
\({p_x} = 1/6\)
Therefore \({p_x} = 1/6\).
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