# Lesson 2

Playing with Probability

• Let’s explore probability

### 2.1: Taking Names

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
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.

### 2.3: Probability Words

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.

Each of the whole numbers from 1 to 25 is written on a slip of paper and placed in a bag.

1. Calculate each probability.
1. $$P(\text{prime})$$
2. $$P(\text{divisible by 3 but not 2})$$
3. $$P(\text{multiple of 5})$$
4. $$P(\text{greater than 20})$$
5. $$P(\text{multiple of 12 and less than 20})$$
2. Use this situation to create two of your own probability questions that give a probability of $$\frac{1}{25}$$ as the answer.
3. Use this situation to create two of your own probability questions that give a probability of $$\frac{3}{25}$$ as the answer.

### Summary

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 $$P(\text{rolling a 2}) = \frac{1}{6}$$.

### Glossary Entries

• chance experiment

A chance experiment is something you can do over and over again, and you don’t know what will happen each time.

For example, each time you spin the spinner, it could land on red, yellow, blue, or green.

• event

An event is a set of one or more outcomes in a chance experiment. For example, if we roll a number cube, there are six possible outcomes.

Examples of events are “rolling a number less than 3,” “rolling an even number,” or “rolling a 5.”

• outcome

An outcome of a chance experiment is one of the things that can happen when you do the experiment. For example, the possible outcomes of tossing a coin are heads and tails.

• probability

The probability of a chance event is a number from 0 to 1 that expresses the likelihood of the event occurring, with 0 meaning it will never occur and 1 meaning it will always occur.

• sample space

The sample space is the list of every possible outcome for a chance experiment.

For example, the sample space for tossing two coins is: