Finding Probability Using Tree Diagrams and Outcome Tables

Tree Diagrams If you flip a coin twice, you can model the possible outcomes using a tree diagram or an outcome table Tree Diagram: Outcome table: H T Flip 1 Flip 2 Simple Event H HH T HT TH TT T

Tree Diagrams if you rolled 1 die and then flipped a coin you have 12 possible outcomes H T (1,H) 1 (1,T) H T (2,H) 2 (2,T) H T (3,H) 3 (3,T) (4,H) H T 4 (4,T) H T (5,H) 5 (5,T) (6,H) H T 6 (6,T)

Multiplicative Principle for Counting How many possible 3-letter words are there? There are 26 possibilities for the first letter There are 26 possibilities for the second letter There are 26 possibilities for the third letter Total possibilities = 26 x 26 x 26 = 17576 How many possible postal codes are there in Canada? Total possibilities = 26 x 10 x 26 x 10 x 26 x 10 = 17 576 000

Calculating probabilities of independent events You roll a die and flip a coin Are they independent or dependent on each other? To get the probability of two independent events occurring, simply multiply the probability of each individual event together P(odd, head) = P(odd) x P(head) = ½ x ½ = ¼

Example 1: tree diagram a sock drawer has a red, a green and a blue sock you pull out one sock, replace it and pull another out a) draw a tree diagram representing the possible outcomes b) what is the probability of drawing 2 red socks? these are independent events R B G

Example 2 a) If you draw a card, replace it and draw another, what is the probability of getting two aces? These are independent events 4/52 x 4/52 b) If you draw an ace and then draw a second card (“without replacement”), what is the probability of two aces? second event depends on first event 4/52 x 3/51

Example 3 - Predicting Outcomes Mr. Xu is playing Texas Hold’Em He finds that he wins 70% of the pots when he does not bluff He also finds that he wins 50% of the pots when he does bluff If there is a 60% chance that Mr. Xu will bluff on his next hand, what are his chances of winning the pot? We will start by creating a tree diagram

Win pot P=0.6 x 0.5 = 0.3 0.5 bluff 0.6 0.5 Lose pot P=0.6 x 0.5 = 0.3 Win pot 0.7 P=0.4 x 0.7 = 0.28 0.4 no bluff 0.3 Lose pot P=0.4 x 0.3 = 0.12

Homework Read the examples on pages 239-244 Try pp. 245 – 249 #2, 3, 5, 7, 9, 12, 13a, 14