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10.3: Probability of Single Events Example 10.3.1 (Calculating Classical Probability Using Combinations): Suppose that as one of the 20 graduate students in the physics department, you have a chance of being selected for one of the three student spots for a conference tip to Cancun. If the names of all the graduate students are put in a hat and three are drawn, what is the probability that you and your two friends, Leonard and Sheldon, end up being chosen? Example 10.3.2 (Calculating Classical Probability Using Permutations): Let’s change the previous example slightly. Suppose that as one of the 20 graduate students in the physics department, you have a chance of being selected for one of the three student spots for a conference tip to Cancun. The names of all the graduate students are put in a hat and three are randomly drawn. However, if your name is drawn first, you get all expenses paid. If you are chosen second, everything is paid for except meals. And if you are chosen third, you must pay for your own meals and hotel. (This means that the department still picks up the tab for the flight and conference fees of the three lucky students, so it’s not a bad deal!) What is the probability that you and your two friends, Leonard and Sheldon, all end up being chosen and that your name is drawn first? Skill Check 10.3.1: Using the scenario from Examples 10.3.1 and 10.3.2, find the probability that you, Sheldon and Leonard are all chose for the department trip and you are the third to be chosen?

Example 10.3.3 (Calculating Classical Probability): The university is putting together a hiring committee of 6 members for the next university provost. The eligible pool of members consists of 8 administrative personnel, 9 faculty and 7 students. a. In how many ways can the committee be formed? b. In how many ways can the committee be formed if there must be 2 members chosen from each of the 3 subgroups? c. Suppose the committee is chosen at random and with no restrictions. What is the probability that each subgroup has 2 members on the committee? Example 10.3.4 (Calculating Classical Probability): An employee is setting up a new password for his in-house messaging account. The password must contain at least 4 but no more than 6 characters using only lowercase letters and the digits 0 through 9, and each character can only be used once. Suppose he lets the computer randomly set his password using all of the restrictions listed. Use the following steps to calculate the probability that the password generated by the computer would contain all the letters in his dog’s name – Bailey. Step 1: Begin by determining how many different passwords are possible given the restrictions. Step 2: Determine the number of possible passwords that contain all the letters in the dog’s name. Step 3: Calculate the probability using the values from Steps 1 and 2.

Example 10.3.5 (Finding the Complement of an Event): Describe the complement for each of the following events: a. Choosing a vowel (a, e, i, o, u) from the English alphabet. b. Choosing a number that doesn’t end in 1 from all positive two-digit whole numbers. c. From a class of 52 students, choosing a student who is over 21 years old. Skill Check 10.3.2: Let the event E be an odd sum when a pair of dice is rolled. For each of the following, determine if the outcome could be in the complement of E. a. A sum greater than 8 b. A sum that is an even number c. A sum that is less than 5 d. A sum that is a multiple of 3 e. All of the above

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