\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Let \(S\) represent the sum of the numbers obtained when rolling two fair six-sided dice: a red die and a blue die . Construct the probability distribution table for \(S\).
\FieldText{1}{The possible values of \(S\) are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. The probabilities are:
  • \(P(S = 2) = P(\{(1,1)\}) = \frac{1}{36}\),
  • \(P(S = 3) = P(\{(1,2), (2,1)\}) = \frac{2}{36}\),
  • \(P(S = 4) = P(\{(1,3), (2,2), (3,1)\}) = \frac{3}{36}\),
  • \(P(S = 5) = P(\{(1,4), (2,3), (3,2), (4,1)\}) = \frac{4}{36}\),
  • \(P(S = 6) = P(\{(1,5), (2,4), (3,3), (4,2), (5,1)\}) = \frac{5}{36}\),
  • \(P(S = 7) = P(\{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\}) = \frac{6}{36}\),
  • \(P(S = 8) = P(\{(2,6), (3,5), (4,4), (5,3), (6,2)\}) = \frac{5}{36}\),
  • \(P(S = 9) = P(\{(3,6), (4,5), (5,4), (6,3)\}) = \frac{4}{36}\),
  • \(P(S = 10) = P(\{(4,6), (5,5), (6,4)\}) = \frac{3}{36}\),
  • \(P(S = 11) = P(\{(5,6), (6,5)\}) = \frac{2}{36}\),
  • \(P(S = 12) = P(\{(6,6)\}) = \frac{1}{36}\).
\(s\) 2 3 4 5 6 7 8 9 10 11 12
\(P(S = s)\) \(\frac{1}{36}\) \(\frac{2}{36}\) \(\frac{3}{36}\) \(\frac{4}{36}\) \(\frac{5}{36}\) \(\frac{6}{36}\) \(\frac{5}{36}\) \(\frac{4}{36}\) \(\frac{3}{36}\) \(\frac{2}{36}\) \(\frac{1}{36}\)
}