Question

A red die has the number 1 on one face and 2 on two faces, and the number 3 on three faces. Two green dice each have the number 6 on one face and the number 5 on five faces. The three dice are rolled together.

(a) Draw a tree diagram.
(b) Using the tree diagram, calculate the probability of obtaining:
(i) 2 on the red die, 5 on the first green die, and 6 on the second green die.
(ii) 3 on the red die and 6 on each of two green dice.
(iii) Exactly 2 sixes.
(iv) A sum of 12.
(v) A sum divisible by 3.

Answer 1

The tree diagram for the dice rolls consists of three layers, representing the red die and two green dice. The probabilities can be calculated by multiplying the probabilities along the corresponding branches of the tree diagram. The probabilities for the given scenarios are: (i) 5/108, (ii) 1/108, (iii) 5/648, (iv) 1/108, and (v) calculated using the tree diagram.

Step-by-step explanation:

a) Tree Diagram

• The tree diagram for this situation would consist of three layers, representing the three dice.
• Start with the red die which has 1, 2, and 3 as possible outcomes.
• For each outcome on the red die, draw two branches for the two green dice, representing the numbers 5 and 6 on each die.
• Label each branch with the corresponding outcome and calculate the probabilities along each path.

b) Probability Calculations

(i) Probability of obtaining 2 on the red die, 5 on the first green die, and 6 on the second green die:

• This probability can be found by multiplying the probabilities along the corresponding branches of the tree diagram.
• The probability is (2/6) x (5/6) x (1/6) = 5/108 or approximately 0.0463.

(ii) Probability of obtaining 3 on the red die and 6 on each of two green dice:

• The probability is (3/6) x (1/6) x (1/6) = 1/108 or approximately 0.0093.

(iii) Probability of obtaining exactly 2 sixes:

• There are 3 ways to obtain exactly 2 sixes: RG RG G, RG GR G, and GR RG G where R represents the red die and G represents a green die.
• The probabilities of each of these outcomes are (2/6) x (1/6) x (5/6) x (1/6) x (1/6) = 5/648 or approximately 0.0077.

(iv) Probability of obtaining a sum of 12:

• The only possible way to obtain a sum of 12 is by obtaining a 3 on the red die and a 6 on each green die.
• The probability is (3/6) x (1/6) x (1/6) = 1/108 or approximately 0.0093.

(v) Probability of obtaining a sum divisible by 3:

• To obtain a sum divisible by 3, the possible combinations are: (3,5,6), (3,6,5), (2,6,6), (6,3,5), (6,5,3), (5,6,3).
• The probabilities of each of these outcomes can be calculated using the tree diagram by multiplying the probabilities along the corresponding branches.