- #1
Beowulf2007
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Homework Statement
Given [tex]S = \{(r,w)| r,w = 1,2,\ldots, 6\}[/tex]
Deduce the following three probability functions.
Probability that the number of eyes are red
(1)[tex]P_{R}(t) = \frac{1}{6}[/tex] for [tex]t \in \{1,2,\ldots 6 \}[/tex]
Probability that the number of eyes are either red or white
(2)[tex]P_{Y}(t) = \frac{13-2t}{36}[/tex] for [tex]t \in \{1,2,\ldots 6 \}[/tex]
Probability that the number of eyes are either red and white
(3)[tex]P_{Z}(t) = \frac{2t-1}{36}[/tex] for [tex]t \in \{1,2,\ldots 6 \}[/tex]
The Attempt at a Solution
My Proof (1):
Since there is 6 sides on each dice the combined space [tex]S = 6 \cdot 6 = 36 [/tex] and since there is 6 sides on each sides of red dice, then
[tex]\frac{6}{36} = \frac{1}{6} = P_{R}(t)[/tex]
My Proof(2):
The Events of throwing the two dice are describe in the schema:
[tex]
\begin{array}{|c| c| c| c| c| c| }
\hline
(1,1) & (1,2) & (1,3) & (1,4) & (1,5) & (1,6)\\
\hline
(2,1) & (2,2) & (2,3) & (2,4) & (2,5) & (2,6)\\
\hline
(3,1) & (3,2) & (3,3) & (3,4) & (3,5) & (3,6)\\
\hline
(4,1) & (4,2) & (4,3) & (4,4) & (4,5) & (4,6)\\
\hline
(5,1) & (5,2) & (5,3) & (5,4) & (5,5) & (5,6)\\
\hline
(6,1) & (6,2) & (5,3) & (5,4) & (6,5) & (6,6)\\
\hline
\end{array}
[/tex]
Thus by in the schema:
[tex]\begin{array}{ccc} P(x = 1) = \frac{11}{36} & P(x = 2) = \frac{9}{36} & P(x = 3) = \frac{7}{36}\\P(x = 4) = \frac{5}{36} & P(x = 5) = \frac{3}{36} & P(x = 6) = \frac{1}{36} \end{array}[/tex]
which can be describe by the function:
[tex]P_{Y}(t) = \frac{13-2t}{36}[/tex] for [tex]t \in \{1,2,\ldots 6 \}[/tex]
Proof(3)
Thus by in the schema:
[tex]\begin{array}{ccc} P(x = 1) = \frac{1}{36} & P(x = 2) = \frac{3}{36} & P(x = 3) = \frac{5}{36}\\P(x = 4) = \frac{7}{36} & P(x = 5) = \frac{9}{36} & P(x = 6) = \frac{11}{36} \end{array}[/tex]
which can be describe by the function:
[tex]P_{Y}(t) = \frac{2t-1}{36}[/tex] for [tex]t \in \{1,2,\ldots 6 \}[/tex]
What You Guys say I have deduced the probability functions correctly?? Am I on the right track??
SIncerely Yours
Beowulf
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