PROBABILITY 12C
TWO WAY TABLES AND TREE DIAGRAMS
TWO WAY TABLES
• When more than one event has to be considered, a visual representation of the sample space
is helpful in calculating the probabilities of various events.
Two-way tables
• A two-way table (sometimes referred to as a lattice diagram) is able to represent two events in
a table form, when the two events occur at the same time
• A two-way table for the experiment of tossing a coin and rolling a die simultaneously is shown
in the following table.
Possibilities
for dice roll
Possibilities for
coin toss
All possible
outcomes/
combinations
WORKED EXAMPLE
• Two dice are rolled. The outcome is the pair of numbers shown.
(a) Show the results on a two-way table.
Dice 1 possibilities
Dice 2 possibilities
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
(b) Calculate the probability of
obtaining an identical ordered pair;
that is,
Pr[(1, 1), (2, 2), (3, 3), (4, 4), (5, 5),
(6, 6)]
Dice 2 possibilities
6 x 6 = 36 total possibilities
How many ordered pairs are there?
6
Therefore, Pr(ordered pair)
6
1
= 36 = 6
Dice 1 possibilities
How many total possibilities are there?
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
TREE DIAGRAMS
• Tree diagrams are very helpful when there are multiple
events; for example, when a coin is tossed twice.
• These are events that happen one after the other, not at
the same time (like two way tables)
• Each stage of a multiple event experiment produces a
part of a tree.
• For example, this tree diagram shows the outcomes of a
coin being tossed once, then tossed again
• To work put the probability of these events occurring,
multiply them together, e.g. Pr(HT) = 0.5 x 0.5 = 0.25
• When you add all the probabilities together, they should
add to one
WORKED EXAMPLE
• Three coins are tossed simultaneously. Draw a tree diagram for the
experiment. Calculate the following probabilities.
• a Pr(3 Heads) b Pr(2 Heads) c Pr(at least 1 Head)
• Three coins are tossed simultaneously. Draw a tree diagram for the
experiment.
Coin 1
H
1
2
1
2
Coin 2
1
2
1
2
T
1
2
1
2
H
T
Coin 3
1
2
1
2
1
2
1
2
H
1
2
1
2
T
1
2
1
2
H
T
H
T
H
T
H
T
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
• Calculate the following probabilities.
a Pr(3 Heads) b Pr(2 Heads) c Pr(at least 1 Head)
1
2
1
2
1
2
(a) Pr(HHH) = x x =
1
8
1
8
1
8
1
8
(b) Pr(two H) = Pr(HHT) + Pr(HTH) + Pr(THH) = + + =
(c) Pr(at least one H) =
How many outcomes have at least one H?
All except TTT
Pr(TTT) =
1
8
1
8
1 – Pr(TTT) = 1 - =
7
8
3
8
WORKED EXAMPLE
• The letters A, B, C and D are written on identical pieces of card and placed
in a box. A letter is drawn at random from the box. Without replacing the
first card, a second one is drawn. Use a tree diagram to find:
a Pr(first letter is A) b Pr(second letter is B) c Pr(both letters are the same).
• Three coins are tossed simultaneously. Draw a tree diagram for the
experiment.
First draw
A
Remember, the card
is removed after
the first draw,
leaving the 3
leftover cards in the
second draw
1
4
1
4
1
4
B
C
1
4
Second draw
1
3
1
3
1
3
1
3 1
3
1
3
1
3
1
31
3
D
1
3 1
3
1
3
B
C
D
A
C
D
A
B
D
A
B
C
AB
AC
AD
BA
BC
BD
CA
CB
CD
DA
DB
DC
1
1
1
1
1
1
1
1
1
1
1
1
Pr(AB) = 4 x 3 = 12
Pr(AC) = 4 x 3 = 12
Pr(AD) = 4 x 3 = 12
Pr(BA) = 4 x 3 = 12
1 1
1
4 3
12
1 1
1
Pr(BD) = 4 x 3 = 12
1 1
1
Pr(CA) = 4 x 3 = 12
1 1
1
Pr(CB) = 4 x 3 = 12
1 1
1
Pr(CD) = 4 x 3 = 12
1 1
1
Pr(DA) = 4 x 3 = 12
1 1
1
Pr(DB) = 4 x 3 = 12
1 1
1
Pr(DC) = 4 x 3 = 12
Pr(BC) = x =
(a) Pr(first letter is A)
= Pr(AA) + Pr(AB) + Pr(AC) + Pr(AD)
=
1
12
=
3
12
+
1
12
=
1
4
+
1
12
(c) Pr(both letters the same)
= Pr(AA) + Pr(BB) + Pr(CC) + Pr(DD)
(b) Pr(second letter B)
= Pr(AB) + Pr(CB) + Pr(DB)
=
1
12
=
3
12
+
1
12
=
1
4
+
1
12
=0
QUESTIONS TO DO
• Exercise 12C p410: 1, 3, 7, 8, 10, 11, 12