Question

two dice are rolled one after another. construct a sample space and determine the probability that the sum of the dots on the dice total a number greater than 2 if the second die is a 1.

the probability that the sum of the dots on the dice is a number greater than 2 if the second die is a 1 is .
(type an integer or a simplified fraction.)

Explanation:

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<reasoning>

Define the reduced sample space

\[
S = \{(1,1), (2,1), (3,1), (4,1), (5,1), (6,1)\}
\]

Identify outcomes satisfying the condition

\[

$$\begin{aligned} &\text{Sums for each outcome in } S:\\ &1+1 = 2 \quad (\text{not } > 2)\\ &2+1 = 3 \quad (> 2)\\ &3+1 = 4 \quad (> 2)\\ &4+1 = 5 \quad (> 2)\\ &5+1 = 6 \quad (> 2)\\ &6+1 = 7 \quad (> 2)\\ &\text{Favorable outcomes: } \{(2,1), (3,1), (4,1), (5,1), (6,1)\} \end{aligned}$$

\]

Calculate the conditional probability

\[
P(\text{Sum} > 2 \mid \text{Second die is } 1) = \frac{5}{6}
\]
</reasoning>

<answer>
The probability that the sum of the dots on the dice is a number greater than 2 if the second die is a 1 is <blank>\(\frac{5}{6}\)</blank>.
</answer>

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Answer:

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<reasoning>

Define the reduced sample space

\[
S = \{(1,1), (2,1), (3,1), (4,1), (5,1), (6,1)\}
\]

Identify outcomes satisfying the condition

\[

$$\begin{aligned} &\text{Sums for each outcome in } S:\\ &1+1 = 2 \quad (\text{not } > 2)\\ &2+1 = 3 \quad (> 2)\\ &3+1 = 4 \quad (> 2)\\ &4+1 = 5 \quad (> 2)\\ &5+1 = 6 \quad (> 2)\\ &6+1 = 7 \quad (> 2)\\ &\text{Favorable outcomes: } \{(2,1), (3,1), (4,1), (5,1), (6,1)\} \end{aligned}$$

\]

Calculate the conditional probability

\[
P(\text{Sum} > 2 \mid \text{Second die is } 1) = \frac{5}{6}
\]
</reasoning>

<answer>
The probability that the sum of the dots on the dice is a number greater than 2 if the second die is a 1 is <blank>\(\frac{5}{6}\)</blank>.
</answer>

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