Homework 7: Independence

1. Two dice, one red and one green, are rolled. Define the events
   • \(A\): The red die lands on a 3, 4, or 5
   • \(B\): The sum of the two dice is 9
   1. Compute \(P(A\mid B)\).
   2. Compute \(P(B\mid A)\).
   3. Are \(A\) and \(B\) independent events? Justify your answer.
2. Recall that three events \(A\), \(B\), and \(C\) are independent if \begin{equation} \begin{split} & P(A\cap B)=P(A)\cdot P(B)\\ & P(B\cap C)=P(B)\cdot P(C)\\ & P(A\cap C)=P(A)\cdot P(C) \end{split} \end{equation} and \begin{equation} P(A\cap B\cap C)=P(A)\cdot P(B)\cdot P(C) \end{equation} Two dice, one red and one green, are rolled. Define the events
   • \(A\): The red die lands on a 1, 2, or 3
   • \(B\): The red die lands on a 3, 4, or 5
   • \(C\): The sum of the two dice is 9
   Do these three events satisfy the first three equations above? Do they satisfy the fourth? Are the three events independent?
3. A roulette wheel has thirty-six numbers colored red or black according to the pattern indicated below: \begin{array}{cccccccccccccccccc} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18\\ R & R & R & R & R & B & B & B & B & R & R & R & R & B & B & B & B & B\\ 36 & 35 & 34 & 33 & 32 & 31 & 30 & 29 & 28 & 27 & 26 & 25 & 24 & 23 & 22 & 21 & 20 & 19\\ \hline \end{array} Define the events
   • \(A\): Red number appears
   • \(B\): Even number appears
   • \(C\): Number is less than or equal to 18
   Do these three events satisfy the first three equations appearing in Problem 2 above? Do they satisfy the fourth? Are the three events independent?