One die and two dice: the table of 36 pairs, tree diagrams, and drawing balls with and without replacement
Statistics Sec 3 Term 1, Unit 3, Lesson 1
GoalWrite the sample space for one die and two dice, and for drawing balls with and without replacement, using tables and tree diagrams.
New words
ordered pair · (first die, second die): (2, 5) is not the same outcome as (5, 2)with replacement · the drawn object is put back before the next drawwithout replacement · the drawn object is not put back, so it cannot appear again
Explanation
One die has the sample space S = {1, 2, 3, 4, 5, 6}, n(S) = 6. Two dice (or one die rolled twice) give ordered pairs (first, second), 6 × 6 = 36 of them. The table lists them all: the row is the first die and the column the second, so the pair (2, 5) is in the row 2 and the column 5. For drawing balls the answer depends on whether the ball is put back. Take a bag with a red (R), a blue (B) and a green (G) ball and draw two. With replacement the ball goes back, so the second draw has the same 3 choices: 3 × 3 = 9 outcomes. Without replacement the second draw has only 2 balls left: 3 × 2 = 6 outcomes, and the same ball cannot appear twice, so (R, R) is not possible. In a tree each branch is a choice, and multiplying the numbers of branches along the levels gives n(S).
1st \ 2nd
1
2
3
4
5
6
1
(1, 1)
(1, 2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
2
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
3
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
4
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
5
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
6
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
Worked examples
1st \ 2nd
1
2
3
4
5
6
1
(1, 1)
(1, 2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
2
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
3
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
4
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
5
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
6
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
Two dice: reading the table. There are 36 cells, so n(S) = 36. The outcome “first die 4, second die 1” is the cell (4, 1) in the row 4, and it is a different cell from (1, 4). The cells with the same number on both dice, (1, 1), (2, 2), …, (6, 6), form the diagonal: 6 cells.
Three balls, with replacement. The bag holds R, B and G, and two balls are drawn, the first being put back. S = {(R, R), (R, B), (R, G), (B, R), (B, B), (B, G), (G, R), (G, B), (G, G)}, n(S) = 3 × 3 = 9.
The same bag, without replacement. S = {(R, B), (R, G), (B, R), (B, G), (G, R), (G, B)}, n(S) = 3 × 2 = 6. The tree has only two branches at the second level. A second example: four cards numbered 1 to 4, two drawn. With replacement n(S) = 4 × 4 = 16; without replacement n(S) = 4 × 3 = 12.
Class activity
Roll one die twice (or two dice) 12 times and write down the pairs. Mark them on the table of 36 cells. Were any cells marked twice? How many cells stay empty, and does that surprise you?
Practice
1
How many outcomes are in the sample space of two dice? Of three dice?
6 × 6 = 36; 6 × 6 × 6 = 216.
2
A bag has 4 cards A, B, C, D. Two are drawn without replacement. Write n(S) and list the outcomes that start with A.
