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(r3 and w2)P (r ≤ 3 and w ≤ 2)P (r ≤ 3)18 (r 3)36 (die pairs) 1 2 P (w ≤ 2)12 (w ≤ 2)36 (die pairs) 1 3 P (r ≤ 3 and w ≤ 2)P (r 3 ∩ w ≤ 2) 1 • 1 2 3 1 6

(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) 2 136 18P (r =2 or w 5)P (r = 2) + P (w 5) – P (r = 2 ∩ w 5) 6 + 12 – 236 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(r2 and w4)P (r 2 and w 4) P (r ≤ 2)12 (r 2)36 (die pairs) 1 3 P (w ≤ 4)24 (w ≤ 4)36 (die pairs) 2 3 P (r 2 and w 4)P (r 2 ∩ w 4) 1 • 2 3 3 2 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(r4 and w=4)P(r 4 and w = 4)P(r 4)18 r 4

36 (die pairs) 1 2 P (w = 4) 6 (w = 4)36 (die pairs) 1 6 P(r 4 and w = 4)P(r 4 ∩ w = 4) 1 • 1 2 6 1 12

(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(r5 and w 2)P (r 5 and w 2)P (r 5 )12 (r 5 36 (die pairs) 1 3 P (w ≤ 2)12 (w ≤ 2)36 (die pairs) 1 3 P (r 5 and w 2)P (r 5 ∩ w 2) 1 • 13 3 1 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)

Make a sample space using a tree diagram showing all the possibilities for boys and girls in a family with three children.

Girl (G) or Boy (B)

G or 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)

(1st 4 in sample space)

4

8

1

2

5) P (at most there is three boys)

8

8

1

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 • 5252 52 52 1 13

16) P (only first card is a 4)__• __• __13 13 13 1• 12 • 1213 13 13 1442197

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 color

6(1 • 2 • 4)

3 5 15

16

75