Independent Events OBJ: Find the probability of independent events.

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Presentation transcript:

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 (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)(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) P (r =2 or w  5) P (r = 2) + P (w  5) – P (r = 2 ∩ w  5) – (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 (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, 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 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)(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 (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, 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 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) (1 st 4 in sample space) ) 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) __ __ __ _ )P (no digit is zero) __ __ __

Draw 3 cards, replacing after each draw. 12) P (only one red) RRR RRB RBR RBB BRR BRB BBR BBB ) P (1 st card is 4) ) P (only first card is a 4) __ __ __

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 = P(R) = 6 = ( 1) (1) ( 2)

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) st 2 nd 3 rd color color color 6(1 2 4)