Learning Resources
Lesson
During this part of the chapter, you will be given the opportunity to use the mathematical concepts that you have learned. The Case Study problem will reflect the general nature of the chapter. This problem requires you to use the key skills, knowledge, principles, processes and concepts presented in this unit. You will be able to challenge these problems and may be requested to do part or all of the work. Consult your teacher.
- Case Study 1: Toast
- Case Study 2: Birthday
- Case Study 3: Rolling Dice
Case Studies: Toast
In this case study you will:
- Apply the skills and processes developed in the unit to perform an experimental investigation
- Analyze the results of the experiment and calculate the corresponding probabilities.
This case study obviously deals with experimental probability. There is no way to get the necessary data other than by using an experiment. Since it may not be practical to have buttered toast in the classroom, the experiment has been performed and the results given in the table. Complete the case study based on this data.
Case Studies: Birthday
In this case study you will demonstrate an ability to:
- apply your understanding of the work completed in the chapter especially with respect to the complement of an event
- apply your understanding of binomial probabilities to solve a problem
Caution: Question (a) is somewhat deceptive. The temptation is to reason as follows: since there are 365 days in a year, the probability that any given student's birthday falls on a particular day (viz. your birthday) is 
. That is a correct statement. The mistake occurs when we multiply this probability by the number in the class. For example, if there are 10 people in class the probability that "someone" has the same birthday as you is NOT 
. It may be close to it but it is not equal to it. You can see this if you take the extreme case, viz. a class of 365. Using this reasoning, the probability "someone" has the same birthday as you would be 
or 1. It would be guaranteed but quite obviously that is not the case. You have to account for the fact that some of the 365 may have the same birthday. That calculation is difficult even for 2 people as indicated in question (b). Think about the problem but you may not be able to find the solution.
For Question (b) it is easier to calculate the probability of the complement and then subtract from 1 to get the probability of the event. Calculate the probability that the second person's birthday is different than the first. For example, since there are 365 days in the year on which the first birthday can fall, there are 364 "other days" on which the second birthday can fall and be different. Thus the probability the second birthday is different is 
. Similarly, the third birthday could fall on any of 363 days and be different. and its probability is 
. Depending on the number of students, these probabilities continue to decrease and the probability of all events occurring can be found using the multiplication principle.
This is the probability all the birthdays are different, call it P(X). The probability that at least two are the same is therefore 1 - P(X).
Case Studies: Rolling Dice
In this case study you will demonstrate an ability to:
- translate problems effectively into smaller parts
- solve problems by applying skills with probability
- make conclusions based on probabilities obtained
Rolling one 6 in four rolls of the die can be modeled as in the Section 6, lesson 2 review this idea before beginning the case study.
Rolling at least one double 6 in 24 throws of a pair of dice can be thought of as the complement of the event rolling no double 6s in 245 throws of a pair of dice, which is 24C0 . As well, note with two dice there are 6 x 6 = 36 different possible outcomes. Consider what will be the probability of obtaining 1 double six and the probability of not getting a double 6. Then use the theoretical model for binomial probabilities.
Activity
Complete case study and extension as directed by your teacher.
Test Yourself
There are no test items for the Case Studies.