MHB Calculating Probability for Random Variables with Two Dice

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The discussion focuses on calculating the probability of specific outcomes when rolling two six-sided dice. The probability space is defined, and the probability function assigns equal likelihood to each outcome. For the random variable X1, which sums the values of the two dice, the correct calculation shows that P(X1=7) equals 1/6, as there are six combinations that yield a sum of 7. Participants also note a correction to the probability space, confirming that the last element should be (6,6) instead of (6,1). The conversation emphasizes clarity in identifying outcomes and calculating probabilities accurately.
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Hey! :o

Two usual six-sided dice are thrown. We are given the probability space $\Omega:=\{(1,1),(1,2),..., (1,6), (2,1),..., (6,1)\}$ and the probability function $ P:Pot (\Omega)\rightarrow [0,1], \ P (\{(a,b)\})=\frac{1}{36}$.
Calculate for the following random variable $X_i:\Omega->R$ the probability $P(X_i=7)$.

a)$X_1 ((a,b))=a+b$

b)$X_2 ((a,b))=2a+1$

c) $X_3 ((a,b))=6+a-b$

Could you give me a hint what I am supposed to do? (Wondering)
 
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mathmari said:
Hey! :o

Two usual six-sided dice are thrown. We are given the probability space $\Omega:=\{(1,1),(1,2),..., (1,6), (2,1),..., (6,1)\}$ and the probability function $ P:Pot (\Omega)\rightarrow [0,1], \ P (\{(a,b)\})=\frac{1}{36}$.
Calculate for the following random variable $X_i:\Omega->R$ the probability $P(X_i=7)$.

a)$X_1 ((a,b))=a+b$

b)$X_2 ((a,b))=2a+1$

c) $X_3 ((a,b))=6+a-b$

Could you give me a hint what I am supposed to do? (Wondering)

Hey mathmari! (Smile)

For the first sub question we're supposed to calculate (since the probability for each individual outcome is the same):
$$P(X_1=7) = P(a+b=7) = \frac{\text{#outcomes such that a+b=7}}{\text{#outcomes in total}}$$
where $a$ is the value of the first die, and $b$ is the value of the second die. (Thinking)
 
I like Serena said:
Hey mathmari! (Smile)

For the first sub question we're supposed to calculate (since the probability for each individual outcome is the same):
$$P(X_1=7) = P(a+b=7) = \frac{\text{#outcomes such that a+b=7}}{\text{#outcomes in total}}$$
where $a$ is the value of the first die, and $b$ is the value of the second die. (Thinking)

So, we get that $$P(X_1=7) = P(a+b=7) = \frac{(1,6), (6,1)}{\#\Omega}=\frac{2}{36}=\frac{1}{18}$$ or not? (Wondering)
 
Don't we have more elements in the numerator than just $(1,6), (6,1)$? (Wondering)

Oh, and shouldn't the last element in $\Omega$ in the problem statement be (6,6) instead of (6,1)?
 
I like Serena said:
Don't we have more elements in the numerator than just $(1,6), (6,1)$? (Wondering)

So, we have at the numerator the elements $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$.

Therefore we get $$P(X_1=7) = P(a+b=7) = \frac{\#\{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\}}{\#\Omega}=\frac{6}{36}=\frac{1}{6}$$ right? (Wondering)
I like Serena said:
Oh, and shouldn't the last element in $\Omega$ in the problem statement be (6,6) instead of (6,1)?

Oh yes (Blush)
 
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Yep. (Nod)
 
I like Serena said:
Yep. (Nod)

Thank you! (Sun)
 
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