 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Solutions for Session 8, Part C

See solutions for Problems: C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11    Problem C1 Two branches of the tree end with one head out of two tosses (HT and TH), and only one branch ends with zero heads (TT). Therefore, it is more likely to get one head than no heads.   Problem C2 Here is the tree diagram for three tosses of a fair coin:    Problem C3

Here is the completed probability table:   Frequency Probability  0 1 1/8 1 3 3/8 2 3 3/8 3 1 1/8    Problem C4

Here is the completed probability table:   Frequency Probability  0 1 1/16 1 4 4/16 2 6 6/16 3 4 4/16 4 1 1/16    Problem C5 No, it would not be feasible to plot the necessary branches. There would be a total of 210 = 1,024 branches to this tree diagram, and it would be far too cumbersome to count all the outcomes.   Problem C6 The fifth row is 1, 5, 10, 10, 5, 1, and the sixth row is 1, 6, 15, 20, 15, 6, 1, as shown below:    Problem C7

Here are the completed probability tables:

Five Tosses   Frequency Probability  0 1 1/32 1 5 5/32 2 10 10/32 3 10 10/32 4 5 5/32 5 1 1/32 Six Tosses   Frequency Probability  0 1 1/64 1 6 6/64 2 15 15/64 3 20 20/64 4 15 15/64 5 6 6/64 6 1 1/64    Problem C8 If you say that P = the number of possible outcomes, and n is the number of tosses, then P = 2n.   Problem C9 The seventh row is 1, 7, 21, 35, 35, 21, 7, 1. The eighth row is 1, 8, 28, 56, 70, 56, 28, 8, 1. The ninth row is 1, 9, 36, 84, 126, 126, 84, 36, 9, 1. The 10th row is 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1. The frequency of five heads in 10 coin tosses is the sixth number in this row, which is 252 (note that it is the center number in the row). Since there are 210 = 1,024 possible outcomes in this row, the probability of getting five heads out of 10 tosses is 252/1,024, or about 24.6%.   Problem C10

 a. The most probable score is two correct. It has a probability of 6/16. b. The least probable scores are zero correct and four correct. Each has a probability of 1/16. c. The probability of getting at least two answers correct is 6/16 + 4/16 + 1/16 = 11/16. d. The probability of getting at least three answers correct is 4/16 + 1/16 = 5/16.   Problem C11 The simplest way to approach this problem is to find the probability of getting less than two correct, then subtracting this from one. The probability of getting less than two correct is 1/1,024 + 10/1,024 = 11/1024, so the alternate probability is 1 - 11/1,024 = (1,024/1,024) - 11/1,024 = 1,013/1,024, or approximately 98.9%.     