\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Let \(P\) represent the product of the numbers obtained when rolling two fair four-sided dice, each numbered 1, 2, 3, 4: a red die and a blue die . Construct the probability distribution table for \(P\).
\FieldText{1}{The possible values of \(P\) are 1, 2, 3, 4, 6, 8, 9, 12, 16. The probabilities are:
  • \(P(P = 1) = P(\{(1,1)\}) = \frac{1}{16}\),
  • \(P(P = 2) = P(\{(1,2), (2,1)\}) = \frac{2}{16}\),
  • \(P(P = 3) = P(\{(1,3), (3,1)\}) = \frac{2}{16}\),
  • \(P(P = 4) = P(\{(1,4), (2,2), (4,1)\}) = \frac{3}{16}\),
  • \(P(P = 6) = P(\{(2,3), (3,2)\}) = \frac{2}{16}\),
  • \(P(P = 8) = P(\{(2,4), (4,2)\}) = \frac{2}{16}\),
  • \(P(P = 9) = P(\{(3,3)\}) = \frac{1}{16}\),
  • \(P(P = 12) = P(\{(3,4), (4,3)\}) = \frac{2}{16}\),
  • \(P(P = 16) = P(\{(4,4)\}) = \frac{1}{16}\).
Complete the table:
\(p\) 1 2 3 4 6 8 9 12 16
\(P(P = p)\) \(\frac{1}{16}\) \(\frac{2}{16}\) \(\frac{2}{16}\) \(\frac{3}{16}\) \(\frac{2}{16}\) \(\frac{2}{16}\) \(\frac{1}{16}\) \(\frac{2}{16}\) \(\frac{1}{16}\)
}