The purpose of the discussion is to help students understand the total number of outcomes in the sample space and the number of outcomes that are in an event.
Direct students’ attention to the reference created using Collect and Display. Tell students, “Elena actually got a 90% on the first quiz and a 95% on the third quiz.” Ask students to share whether they believe this is impossible, unlikely, or expected and to share their reasoning. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. (For example, the display may have “there are more wrong answers on the first quiz than the third” already on it and can be updated with the more precise phrase “the sample space for each question on the first quiz is has more outcomes than the sample space for the third quiz.”)
If it doesn’t come up, define the terms “chance experiment,” “outcome,” “sample space,” and “event.” A chance experiment is something that can happen for which the result is not already known. An outcome is a possible result for each chance experiment. The collection of all possible outcomes is called the sample space. An event is a group of outcomes from the sample space.
Here are some questions for discussion.
- “What is an example of a chance experiment in this activity?” (Each quiz could be a chance experiment. Each question on a quiz could also be considered a chance experiment.)
- “For each question on the second quiz, what is the sample space? How many outcomes are in the sample space?” (It consists of 4 outcomes: A, B, C, and D).
- “What event is Elena concerned with in this task?” (Elena is concerned with the event that she will get the question correct.)
- “How many outcomes are in the event of getting a question correct on the second quiz before the teacher realized the mistake?” (For each question, there was a single correct choice for the question, so there is 1 outcome in the event.)
- “How many outcomes are in the event of getting a question correct on the second quiz after the teacher realized the mistake?” (For each question, there are 2 outcomes in the event.
Ask students if anyone can recall the meaning of the term “probability.” If necessary, tell students, “A numerical value that represents the chance of an event occurring is called the probability of that event. Probabilities are given as either numerical values between 0 and 1 or as a percentage.”
Here are some questions for discussion.
- “On the second quiz, what was the original probability of Elena getting a correct answer for each question?” (Elena originally had a probability of 0.25 of getting the answer correct for each question.)
- “What did the probability change to when the teacher’s mistake was realized?” (It was raised to 0.5 when the teacher’s mistake was realized. Similarly, Elena had a 50% probability of getting the answer correct for each question on the third quiz.)
If time permits, discuss questions such as these:
- “What is the sample space for each question on quiz three?” (It consists of 2 outcomes: True and False.)
- “If there were 5 choices for each multiple choice question and only one of the choices was correct, what is the probability that Elena would get one right by guessing?” (0.2 or 20%)
- “What is an example of an event and a sample space that you have encountered previously when studying probability?” (One example is to pick a whole number between 1 and 10 and find the probability that it is odd. The sample space is {1,2,3,4,5,6,7,8,9,10} and the event is {1,3,5,7,9}.)
