Independent Events
OBJ: • Find the probability of
independent events
DEF:  Independent Events
2 events that do not depend on each
other
(one event occurring has no relationship
to the other event occurring)
EX:  In a throw of a red die, r, and
a white die, w, find: P(r3 and w2)
P (r ≤ 3 and w ≤ 2)
P (r ≤ 3)
18
36
1
2
P (w ≤ 2)
12
36
1
3
P (r ≤ 3 and w ≤ 2)
P (r  3 ∩ w ≤ 2)
1•1
2 3
1
6
(r  3)
(die pairs)
(w ≤ 2)
(die pairs)
(1, 1)
(1,2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
EX:  In a throw of a red die, r, and
a white die, w, find: P (r=2 or w  5)
P (r = 2 or w  5)
P (r = 2)
6
1
(r = 2)
36
6
(die pairs)
P (w  5)
12
1
(w  5)
36
3
(die pairs)
P (r = 2 and w  5)
P (r = 2 ∩ w  5)
2
1
36
18
P (r =2 or w  5)
P (r = 2) + P (w  5) – P (r = 2 ∩ w  5)
6 + 12 – 2
36 36 36
16
36
4
9
(1, 1)
(1,2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
EX:  In a throw of a red die, r, and
a white die, w, find: P(r2 and w4)
P (r  2 and w  4)
P (r ≤ 2)
12
36
1
3
P (w ≤ 4)
24
36
2
3
P (r  2 and w  4)
P (r  2 ∩ w  4)
1•2
3 3
2
9
(r  2)
(die pairs)
(w ≤ 4)
(die pairs)
(1, 1)
(1,2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
EX:  In a throw of a red die, r, and
a white die, w, find: P(r4 and w=4)
P(r  4 and w = 4)
P(r  4)
18
36
1
2
P (w = 4)
6
36
1
6
P(r  4 and w = 4)
P(r  4 ∩ w = 4)
1•1
2 6
1
12
r4
(die pairs)
(w = 4)
(die pairs)
(1, 1)
(1,2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
EX:  In a throw of a red die, r, and
a white die, w, find:P(r5 and w 2)
P (r  5 and w  2)
P (r  5 )
12
36
1
3
P (w ≤ 2)
12
36
1
3
P (r  5 and w  2)
P (r  5 ∩ w  2)
1•1
3 3
1
9
(r  5
(die pairs)
(w ≤ 2)
(die pairs)
(1, 1)
(1,2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
Make a sample space using a tree diagram
showing all the possibilities for boys and girls
in a family with three children.
Girl (G)
G
or
or
B
Boy (B)
G
or
B
G or B G or B G or B G or B
GGG
GGB
GBG
GBB
BGG
BGB
BBG
BBB
EX:  Find:
GGG, GGB, GBG, GBB, BGG, BGB, BBG, BBB
 P(3 boys)
 P(BBB)
·
·

1
8
 P( 2 boys and a girl)
(BBG or BGB or GBB)
·· + ·· + ··
3 (··)
3
8
EX:  Find:
GGG, GGB, GBG, GBB, BGG, BGB, BBG, BBB
3) P (oldest child is a girl) 5) P (at most there is
three boys)
(1st 4 in sample space)
8
4
8
8
1
1
2
A lottery game consists of choosing a
sequence of 3 digits. Repetition of digits is
allowed, and any digit may be in any position.
6)P (654)
__• __• __
10 10 10
1• 1• 1
10 10 10
1_
1000
7)P (no digit is zero)
__• __• __
10 10 10
9• 9• 9
10 10 10
729
1000
Draw 3 cards, replacing after each
draw.
12) P (only one red)
RRR
RRB
RBR
RBB
BRR
BRB
BBR
BBB
3
8
15) P (1st card is 4)
4 • 52 • 52
52 52 52
1
13
16) P (only first card is a 4)
__• __• __
13 13 13
1• 12 • 12
13 13 13
144
2197
A jar contains 5 blue marbles, 6 red marbles, and 4
yellow marbles. Draw 3 marbles, one at a time,
replacing each marble after it’s drawn.
18.P( two blue and one red)
RRR, RRB, RRB, RBB, BRR, BRB, BBR, BBB
P (BBR, BRB, RBB)
P(B) = 5 = 1
15 3
P(R) = 6 = 2
15 5
3 ( 1) (1) ( 2)
3 3 5
2
15
A jar contains 5 blue marbles, 6 red marbles,
and 4 yellow marbles. Draw 3 marbles, one at
a time, replacing each marble after it’s drawn.
20. P (one of each
color)
5 • 6• 4
15 15 15
1 • 2• 4
3 5 15
3 • 2 • 1
1st
2nd
3rd
color color
6(1 • 2 • 4)
3 5 15
16
75
color