14. extension: mr. silvas surprises his students with a probability challenge that will determine whether…

14. extension: mr. silvas surprises his students with a probability challenge that will determine whether they will take a quiz. he puts three cubes in each of three paper bags: a red cube, a white cube, and a blue cube. he divides the class into four groups of students and tells them that each group will draw one cube from each bag. each group is challenged to come up with a rule to determine whether they will take the quiz based on if their selection of cubes matches or does not match a criteria they identify in advance. two groups establish a rule for a cube combination that leads to the outcome of taking the test, and the other two groups establish a rule for a cube combination that leads to the outcome of not taking the test.\na. draw a tree diagram to show the sample space.

14. extension: mr. silvas surprises his students with a probability challenge that will determine whether they will take a quiz. he puts three cubes in each of three paper bags: a red cube, a white cube, and a blue cube. he divides the class into four groups of students and tells them that each group will draw one cube from each bag. each group is challenged to come up with a rule to determine whether they will take the quiz based on if their selection of cubes matches or does not match a criteria they identify in advance. two groups establish a rule for a cube combination that leads to the outcome of taking the test, and the other two groups establish a rule for a cube combination that leads to the outcome of not taking the test.\na. draw a tree diagram to show the sample space.

Answer

Explanation:

Step1: First bag (Red bag)

The first bag has 3 cubes: red (R), red (R), white (W).

Step2: Second bag (White bag)

The second bag has 3 cubes: blue (B), blue (B), white (W).

Step3: Third bag (Blue bag)

The third bag has 3 cubes: blue (B), blue (B), white (W).

We start with the first - bag. For each cube in the first bag, we branch out for the cubes in the second bag, and then for each of those combinations, we branch out for the cubes in the third bag.

If the first - bag cube is (R_1) (first red cube in the first bag):

  • If the second - bag cube is (B_1) (first blue cube in the second bag), and the third - bag cube is (B_1) (first blue cube in the third bag), the combination is ((R_1,B_1,B_1))
  • If the second - bag cube is (B_1) (first blue cube in the second bag), and the third - bag cube is (B_2) (second blue cube in the third bag), the combination is ((R_1,B_1,B_2))
  • If the second - bag cube is (B_1) (first blue cube in the second bag), and the third - bag cube is (W) (white cube in the third bag), the combination is ((R_1,B_1,W))
  • If the second - bag cube is (B_2) (second blue cube in the second bag), and the third - bag cube is (B_1) (first blue cube in the third bag), the combination is ((R_1,B_2,B_1))
  • If the second - bag cube is (B_2) (second blue cube in the second bag), and the third - bag cube is (B_2) (second blue cube in the third bag), the combination is ((R_1,B_2,B_2))
  • If the second - bag cube is (B_2) (second blue cube in the second bag), and the third - bag cube is (W) (white cube in the third bag), the combination is ((R_1,B_2,W))
  • If the second - bag cube is (W) (white cube in the second bag), and the third - bag cube is (B_1) (first blue cube in the third bag), the combination is ((R_1,W,B_1))
  • If the second - bag cube is (W) (white cube in the second bag), and the third - bag cube is (B_2) (second blue cube in the third bag), the combination is ((R_1,W,B_2))
  • If the second - bag cube is (W) (white cube in the second bag), and the third - bag cube is (W) (white cube in the third bag), the combination is ((R_1,W,W))

If the first - bag cube is (R_2) (second red cube in the first bag):

  • If the second - bag cube is (B_1) (first blue cube in the second bag), and the third - bag cube is (B_1) (first blue cube in the third bag), the combination is ((R_2,B_1,B_1))
  • If the second - bag cube is (B_1) (first blue cube in the second bag), and the third - bag cube is (B_2) (second blue cube in the third bag), the combination is ((R_2,B_1,B_2))
  • If the second - bag cube is (B_1) (first blue cube in the second bag), and the third - bag cube is (W) (white cube in the third bag), the combination is ((R_2,B_1,W))
  • If the second - bag cube is (B_2) (second blue cube in the second bag), and the third - bag cube is (B_1) (first blue cube in the third bag), the combination is ((R_2,B_2,B_1))
  • If the second - bag cube is (B_2) (second blue cube in the second bag), and the third - bag cube is (B_2) (second blue cube in the third bag), the combination is ((R_2,B_2,B_2))
  • If the second - bag cube is (B_2) (second blue cube in the second bag), and the third - bag cube is (W) (white cube in the third bag), the combination is ((R_2,B_2,W))
  • If the second - bag cube is (W) (white cube in the second bag), and the third - bag cube is (B_1) (first blue cube in the third bag), the combination is ((R_2,W,B_1))
  • If the second - bag cube is (W) (white cube in the second bag), and the third - bag cube is (B_2) (second blue cube in the third bag), the combination is ((R_2,W,B_2))
  • If the second - bag cube is (W) (white cube in the second bag), and the third - bag cube is (W) (white cube in the third bag), the combination is ((R_2,W,W))

If the first - bag cube is (W) (white cube in the first bag):

  • If the second - bag cube is (B_1) (first blue cube in the second bag), and the third - bag cube is (B_1) (first blue cube in the third bag), the combination is ((W,B_1,B_1))
  • If the second - bag cube is (B_1) (first blue cube in the second bag), and the third - bag cube is (B_2) (second blue cube in the third bag), the combination is ((W,B_1,B_2))
  • If the second - bag cube is (B_1) (first blue cube in the second bag), and the third - bag cube is (W) (white cube in the third bag), the combination is ((W,B_1,W))
  • If the second - bag cube is (B_2) (second blue cube in the second bag), and the third - bag cube is (B_1) (first blue cube in the third bag), the combination is ((W,B_2,B_1))
  • If the second - bag cube is (B_2) (second blue cube in the second bag), and the third - bag cube is (B_2) (second blue cube in the third bag), the combination is ((W,B_2,B_2))
  • If the second - bag cube is (B_2) (second blue cube in the second bag), and the third - bag cube is (W) (white cube in the third bag), the combination is ((W,B_2,W))
  • If the second - bag cube is (W) (white cube in the second bag), and the third - bag cube is (B_1) (first blue cube in the third bag), the combination is ((W,W,B_1))
  • If the second - bag cube is (W) (white cube in the second bag), and the third - bag cube is (B_2) (second blue cube in the third bag), the combination is ((W,W,B_2))
  • If the second - bag cube is (W) (white cube in the second bag), and the third - bag cube is (W) (white cube in the third bag), the combination is ((W,W,W))

Answer:

The sample space consists of 27 possible combinations: ((R_1,B_1,B_1),(R_1,B_1,B_2),(R_1,B_1,W),(R_1,B_2,B_1),(R_1,B_2,B_2),(R_1,B_2,W),(R_1,W,B_1),(R_1,W,B_2),(R_1,W,W),(R_2,B_1,B_1),(R_2,B_1,B_2),(R_2,B_1,W),(R_2,B_2,B_1),(R_2,B_2,B_2),(R_2,B_2,W),(R_2,W,B_1),(R_2,W,B_2),(R_2,W,W),(W,B_1,B_1),(W,B_1,B_2),(W,B_1,W),(W,B_2,B_1),(W,B_2,B_2),(W,B_2,W),(W,W,B_1),(W,W,B_2),(W,W,W))