2.1 The Fundamental Counting Principle, Study notes of Probability and Statistics

Download 2.1 The Fundamental Counting Principle and more Study notes Probability and Statistics in PDF only on Docsity! Math 3201 2.1 The Fundamental Counting Principle The Fundamental Counting Principle is a way of determining the number of possible ways that we can perform two or more operations together. If operations were being performed independent of each other instead of together, then we would NOT use the Fundamental Counting Principle. In Grade 7, you learned some counting methods, such as using tree diagrams and tables. We will solve the next example using tree diagrams, and then we will learn how to solve it using the fundamental counting principle. Example 1: The school cafeteria restaurant offers a lunch combo for $6 where a person can order: 1 sandwich: chicken, turkey, grilled cheese 1 side: fruit, yogurt, soup 1 drink: juice, milk Draw a tree diagram or create a table to determine the possible lunch combos. Suppose we added and extra type of sandwich, an extra type of side dish, and an extra type of drink. What impact would this have on our tree diagram? Would it still be feasible to use a tree diagram? Limitation of Tree Diagrams: In cases where there are a large number of operations being performed together, it is not feasible to construct a tree diagram. It is more reasonable to use the Fundamental Counting Principle to determine the number of ways of performing all the tasks together. Fundamental Counting Principle # of ways of performing tasks together = (# ways to perform 1st task) × (# ways to perform 2nd task) × (# ways to perform 3rd task) × ... Let’s take a look at the lunch combination example using the fundamental counting principle. Note: The Fundamental Counting Principle can only be used when each of the given tasks are being performed together. From our example, there were three tasks: sandwich, side dish and drink. A lunch combination consisted of ALL three tasks. That is, it could NOT include just a sandwich or just a side dish or just a drink. In cases where a student only bought one item, just a sandwich for example, we would not be able to use the fundamental counting principle. We will discuss this in more detail shortly. Example 3: How many possibly outcomes will there be if you flip a coin and roll a die? Example 4: How many possible outcomes will there be if you flip a coin or roll a die? Example 5: A buffet offers 5 different salads, 10 different entrees, 8 different desserts and 6 different beverages. In how many different ways can you choose a salad, an entree, a dessert, and a beverage? Problems With Restrictions/Conditions Some of the examples that we did already that did not allow repetition of letters or numbers in a pattern are said to have restrictions. This changes the numbers for the various tasks. For example, on license plates if repetition was not permitted, there would be 26 possibilities for the first letter, but only 25 for the second, and 24 for the third. Examples 6: How many three digit numbers can you make using the digits 1, 2, 3, 4 and 5 if: (A) repetition of digits is allowed? (B) repetition of digits is not allowed? Examples 7: In how many ways can a teacher seat five boys and three girls in a row of eight seats if a girl must be seated at the each end of the row? (A) Are there any restrictions for seating girls and boys? (B) Why should you fill the girls seats first? (C) How many choices are there for seat 1 if a girl must sit in that seat? (D) How many girls remain to sit in seat 8? (E) How many choices of boys and girls remain to sit in each of seats 2 through 7? (F) Which mathematical operation should you use to determine the total number of arrangements? (G) What is the total number of possible outcomes? Textbook Questions: page 73 - 75 #2 ,3, 6, 7, 8, 9, 14, 16(a)