Lesson 2

Your teacher will give your group a bag containing slips of paper with names on them. It is important not to open the bag to read the slips at any time. It is important to record your group's data in writing.

Every time a student in the class is notably helpful, a teacher puts their name on a slip of paper and puts it into a bag. If the same student is helpful more than once, their name can be entered multiple times. At the end of the month, the teacher draws several names for prizes. Follow these steps to collect data about the names in the bag:

1. Shake the bag, then draw out only 1 slip of paper.
2. Read the name you drew out loud so that everyone in the group can record the name.
3. Return the slip of paper to the bag and pass the bag to the next person in the group.
4. Repeat these steps until each person in the group has had a chance to draw at least 3 names.

Use the data your group collected in the warm-up to answer the questions.

1. Based on the data you collected, estimate the probability of drawing each of these names from your bag. Explain or show your reasoning.
   1. Clare
   2. Lin
   3. Priya
   4. Elena
   5. Jada
   6. Han
   7. Andre
   8. Diego
   9. Noah
2. There are 15 slips of paper in the bag. What names do you think are written on the slips? Explain your reasoning.
3. If you are allowed to keep going around the group, drawing names and replacing them until you had 100 names drawn, how do you think that affects your understanding of what is in the bag?
4. The next month, the bag contains 15 slips as well. Lin’s name is included 5 times, Clare’s name 4 times, Han’s name 3 times, Diego’s name 2 times, and Jada’s name 1 time. The teacher draws names one at a time, replacing them each time. What might the teacher’s list of names drawn look like if she draws 10 times? Is this the only list of names drawn that is possible? Explain your reasoning.

Take turns with your partner coming up with words that have the probabilities given when selecting a letter at random from the word. Each person should try to come up with one word for each situation.

1. \(P(\text{vowel}) = \frac{1}{3}\). \(P(\text{consonant}) = \frac{2}{3}\).
2. \(P(\text{vowel}) = \frac{2}{3}\). \(P(\text{consonant}) = \frac{1}{3}\).
3. \(P(\text{vowel}) = 0.5\). \(P(\text{T}) = \frac{1}{4}\).
4. \(P(\text{S}) = 0.5\). \(P(\text{vowel}) = 0.25\).
5. Think of a word and give your partner at least 2 clues about the word using probability of certain letters or types of letter.

Some probabilities are estimated by doing an experiment, or sometimes simulating the experiment many times and collecting data about how often outcomes come up. For example, a radio show holds a contest in which callers are entered for a chance to win a ticket to a concert in town. The probability of each caller winning is estimated by considering previous similar contests and comparing the number of callers to the number of ticket winners. If a previous contest had 327 callers and 5 ticket winners, then the probability of winning a ticket can be written:

\(P(\text{winning a ticket}) = \frac{5}{327}\) or \(P(\text{winning a ticket}) \approx 0.015\)

Which means that each caller has about a 1.5% chance of winning a ticket to the concert.

Other probabilities can be determined by recognizing the expected relative likelihood of outcomes among all possible outcomes. For example, we know that the probability of rolling a 2 on a standard number cube is \(\frac{1}{6}\) since there are 6 equally likely outcomes in the sample space for each roll and the event of rolling a 2 is one of those outcomes. This can be written as \(P(\text{rolling a 2}) = \frac{1}{6}\).