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Instead of using the values [tex][tex]$\{1,2,3,4,5,6\}$[/tex][/tex] on dice, suppose a pair of dice have the following:

- Die 1: [tex][tex]$\{1,2,2,3,3,4\}$[/tex][/tex]
- Die 2: [tex][tex]$\{1,3,4,5,6,8\}$[/tex][/tex]

Find the probability of rolling a sum of 2 with these dice.

[tex] P\left(D_1+D_2=2\right) = \square [/tex]



Answer :

GinnyAnswer
To find the probability of rolling a sum of 2 with the given pair of dice, we will follow these steps carefully:

1. Identify the Dice Values:
- Die 1 has the values [tex]\(\{1, 2, 2, 3, 3, 4\}\)[/tex].
- Die 2 has the values [tex]\(\{1, 3, 4, 5, 6, 8\}\)[/tex].

2. Calculate the Total Number of Possible Outcomes:
- Each die has 6 faces.
- The total number of possible outcomes when rolling the two dice is calculated by multiplying the number of faces on each die:
[tex]\[ \text{Total possible outcomes} = 6 \times 6 = 36. \][/tex]

3. Identify the Favorable Outcomes for Rolling a Sum of 2:
- We need to count the number of outcomes where the sum of the values on the two dice equals 2.
- To find these favorable outcomes, we examine all possible combinations of values from Die 1 and Die 2.

4. List the Favorable Outcomes:
- We will look at each value from Die 1 and pair it with each value from Die 2 to see which combinations sum to 2.
- The only way to get a sum of 2 is if Die 1 rolls a 1 and Die 2 rolls a 1.

We check:
- [tex]\(1 \)[/tex] (from Die 1) + [tex]\(1 \)[/tex] (from Die 2) [tex]\(= 2\)[/tex].

No other combination provides a sum of 2. Thus, there is only 1 favorable outcome.

5. Calculate the Probability:
- The probability [tex]\( P(D_1 + D_2 = 2) \)[/tex] is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:
[tex]\[ P(D_1 + D_2 = 2) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{1}{36}. \][/tex]

Therefore, the probability of rolling a sum of 2 with these dice is:
[tex]\[ \boxed{0.027777777777777776} \][/tex]

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