Are these independent or dependent events? Review: Mini-Quiz Independent versus Dependent Events Are these independent or dependent events? Tossing two dice and getting a 3 on both of them. You have a bag of marbles: 4 blue, 6 green, and 10 red. Find the probability of choosing a red marble and then a green marble. Choose one marble out of the bag, look at it, then put it back. Then you choose another marble. You have a basket of socks. You need to find the probability of pulling out a green sock and its matching green sock without putting the first sock back. You pick the letter R from a bag containing all the letters of the alphabet. You do not put the R back in the bag before you pick another letter. Slide 1

Class Greeting

Objective: The student will be able to use possibility diagrams and tree diagrams to solve probability problems involving combined events.

Combined Events

Possibility Diagrams and Tree Diagrams Possibility diagrams and tree diagrams are used to list all possible outcomes of a sample space in a systematic and effective manner. These diagrams are useful for finding the probabilities of combined events.

An example of possibility diagrams Two coins are tossed together. The possibility diagram below shows all the possible outcomes: 2nd coin H T H T S = { } Each represents an outcome. HH, HT, TH, TT Example: P(getting 2 heads) P(HH) = 1st coin

An example of tree diagrams Two coins are tossed together. The tree diagram below shows all the possible outcomes: T H 1st coin ½ 2nd coin Outcome HH HT TH TT Probability P(HH) = ½  ½ = ¼ P(HT) = ½  ½ = ¼ P(TH) = ½  ½ = ¼ P(TT) = ½  ½ = ¼ Each outcome is obtained by tracing along a branch from left to right. The probability of each outcome is obtained by multiplying the probabilities along the respective branch. The total probability of all possible outcomes is ¼+¼+¼+¼ = 1.

Sample Question Question: A box contains three cards numbered 1, 3, 5. A second box contains four cards numbered 2, 3, 4, 5. A card is chosen at random from each box. (a) Show all the possible outcomes of the experiment using a possibility diagram or a tree diagram. (b) Calculate the probability that (i) the numbers on the cards are the same, (ii) the numbers on the cards are odd, (iii) the sum of the two numbers on the cards is more than 7.

Solution to Sample Question (a) Using a possibility diagram: 2nd box 1st box 1 2 3 5 4 S = { (1,2), (1,3), (1,4), (1,5), (3,2), (3,3), (3,4), (3,5), (5,2), (5,3), (5,4), (5,5) } n(S) = 12 (b)(i) P(both numbers are the same) = (b)(ii) P(both numbers are odd) = (b)(iii) P(sum > 7) =

Solution to Sample Question (a) Using a tree diagram: (b)(i) P(both numbers are the same) = P[(3,3) or (5,5)] = 2nd box 1 3 5 2 4 1st box (b)(ii) P(both numbers are odd) = P[(1,3) or (1,5) or (3,3) or (3,5) or (5,3) or (5,5)] = (b)(iii) P(sum > 7) = P[(3,5) or (5,3) or (5,4) or (5,5)] =

Lesson Summary: Objective: The student will be able to use possibility diagrams and tree diagrams to solve probability problems involving combined events.

Preview of the Next Lesson: Objective: The student will be able to calculate the Probability of A and B and the Probability of A or B.

Homework Statistics HW 10 and HW 7