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4.5 Probability using Tree Diagrams & Outcome Tables Tree diagrams and outcome tables provide strategies to list outcomes of particular events Allow tracking of sequences to compute probability of a particular compound event 4.5 Probability using Tree Diagrams & Outcome Tables TREE DIAGRAM AND MULTIPLICATIVE PRINCIPLE Used to represent the outcomes of an experiment that are the result of a sequence of simpler experiments Assume that an outcome for each experiment has no influence on the outcome of any other experiment Total number of outcomes is the product of the possible outcomes at each step in the sequence. If a is selected from A and b from B, n((a,b))=n(A) x n(B) 4.5 Probability using Tree Diagrams & Outcome Tables Consider the experiment where you flip a fair coin and then roll a 4 sided die. Draw the tree diagram & list the outcomes. 1 H,1 2 H,2 H 3 H,3 4 H,4 1 T,1 2 T,2 T 3 T,3 4 T,4 4.5 Probability using Tree Diagrams & Outcome Tables Consider the previous example. Would writing the ordered pairs as (roll, flip) change the effect on the total outcomes? How would the tree differ? What is the probability of flipping a head and rolling an odd number? P(H, odd) = 2/8 = 1/4 4.5 Probability using Tree Diagrams & Outcome Tables INEPENDENT AND DEPENDENT EVENTS In the coin toss/die roll experiment: n n(heads, odd rolls) rolls) P H odd P(( H,,odd )) outcomes ) n(outcomes) n(heads) nodd rolls n odd rolls n(coin tosses) ndie rolls n(coin tosses) n die rolls nheads nodd rolls nheads nodd rolls ncoin tosses ndie rolls ncoin tosses ndie rolls P(heads) P(odd P(heads) P(odd )) 1 2 1 12 1 24 4 2 4 4 4.5 Probability using Tree Diagrams & Outcome Tables Does the outcome of the coin toss have any effect on the outcome of the die roll? No! They are independent events. Consider the conditional probability P(H|odd). 4.5 Probability using Tree Diagrams & Outcome Tables P ( H odd ) P( H | odd ) P (odd ) 2 8 1 2 1 2 But we know that P(H)=1/2 Therefore, P(H|odd)=P(H) because the events are independent! 4.5 Probability using Tree Diagrams & Outcome Tables Independent event – events A & B are said to be independent if the occurrence of one event does not change the probability of the occurrence of the other event To find the probability of several things happening in succession, multiply the probabilities of the individual happenings. 4.5 Probability using Tree Diagrams & Outcome Tables 4.5 Probability using Tree Diagrams & Outcome Tables Ex.1 A girl is told by her boyfriend that she is “one in a billion”. She has a dimple in her chin, probability 1/100, eyes of different colours, probability 1/1000, and is absolutely crazy about mathematics, probability 1/10,000. Are these events independent or dependent? Show why the girl is “one in a billion”. 4.5 Probability using Tree Diagrams & Outcome Tables Ex 2 Sean Connery 007 once bet on the number 17 three times in succession at a roulette game in the St. Vincent casino. All three times the number 17 came up and Mr. Connery won $20,000. Are the events that 17 came up 3 times in a row independent or dependent? The wheel had 37 compartments, numbered 0 through 36. What is the probability that the number 17 would come up on: A single spin Two successive spins Three successive spins 4.5 Probability using Tree Diagrams & Outcome Tables Ex 3 A drawer has 6 blue socks and 10 red socks. What is the probability of: Getting two blue socks Getting two red socks Getting one sock of each colour 4.5 Probability using Tree Diagrams & Outcome Tables Ex 3 – Solution Draw a tree diagram 9/15 R RR P(RR)=10/16*9/15=3/8 R 10/16 6/15 B RB P(RB)=10/16*6/15=1/4 ¼+ ¼ = ½ 10/15 R 6/16 BR P(BR)=6/16*10/15=1/4 B 5/15 B BB P(BB)=6/16*5/15=1/8 4.5 Probability using Tree Diagrams & Outcome Tables Home Entertainment P 245 #1-4, 5ab, 6, 8, 11, 12

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posted: | 4/14/2011 |

language: | English |

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