To find the probability of each possible sum when rolling a pair of six-sided dice, let's first identify the total number of possible outcomes. Each die has 6 faces, so the total number of outcomes when rolling two dice is:
[tex]\[ 6 \times 6 = 36 \][/tex]
Next, let's analyze the frequency of each possible sum from 2 to 12, as derived from the table where the sums of each possible roll were provided.
We find these frequencies:
- The sum 2 can occur in 1 way (1+1).
- The sum 3 can occur in 2 ways (1+2, 2+1).
- The sum 4 can occur in 3 ways (1+3, 2+2, 3+1).
- The sum 5 can occur in 4 ways (1+4, 2+3, 3+2, 4+1).
- The sum 6 can occur in 5 ways (1+5, 2+4, 3+3, 4+2, 5+1).
- The sum 7 can occur in 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1).
- The sum 8 can occur in 5 ways (2+6, 3+5, 4+4, 5+3, 6+2).
- The sum 9 can occur in 4 ways (3+6, 4+5, 5+4, 6+3).
- The sum 10 can occur in 3 ways (4+6, 5+5, 6+4).
- The sum 11 can occur in 2 ways (5+6, 6+5).
- The sum 12 can occur in 1 way (6+6).
To find the probability of each sum, we divide the frequency of each sum by the total number of outcomes (36). Here are the details of each probability, calculated as the frequency divided by 36:
- Probability of sum 2: [tex]\( \frac{1}{36} \approx 0.02778 \)[/tex]
- Probability of sum 3: [tex]\( \frac{2}{36} = \frac{1}{18} \approx 0.05556 \)[/tex]
- Probability of sum 4: [tex]\( \frac{3}{36} = \frac{1}{12} \approx 0.08333 \)[/tex]
- Probability of sum 5: [tex]\( \frac{4}{36} = \frac{1}{9} \approx 0.11111 \)[/tex]
- Probability of sum 6: [tex]\( \frac{5}{36} \approx 0.13889 \)[/tex]
- Probability of sum 7: [tex]\( \frac{6}{36} = \frac{1}{6} \approx 0.16667 \)[/tex]
- Probability of sum 8: [tex]\( \frac{5}{36} \approx 0.13889 \)[/tex]
- Probability of sum 9: [tex]\( \frac{4}{36} = \frac{1}{9} \approx 0.11111 \)[/tex]
- Probability of sum 10: [tex]\( \frac{3}{36} = \frac{1}{12} \approx 0.08333 \)[/tex]
- Probability of sum 11: [tex]\( \frac{2}{36} = \frac{1}{18} \approx 0.05556 \)[/tex]
- Probability of sum 12: [tex]\( \frac{1}{36} \approx 0.02778 \)[/tex]
Therefore, the completed table with the probabilities is:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Sum & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
Probability & 0.02778 & 0.05556 & 0.08333 & 0.11111 & 0.13889 & 0.16667 & 0.13889 & 0.11111 & 0.08333 & 0.05556 & 0.02778 \\
\hline
\end{tabular}
