How Likely is it? mathematical and problem solving goals:
How Likely is it? was created to help students:
- Become acquainted with probability informally through experiments
- Understand that 1)rohahilities are usefttl for predicting what will happen over the long run
- Understand that prol)abilities are useliil for making decisions
- Understand that there are two ways to obtain probabilities: by gathering data from experiments (experimental probability) and by analyzing the possible equally likely outcomes (theoretical probability)
- Understand the concepts of equally likely and unequally likely
- Understand the relationship between experimental and theoretical probabilities: experimental probabilities are better estimates of theoretical probabilities when they are based on larger numbers of trials
- Determine and critically interpret statements of probability
- Develop strategies for finding both experimental and theoretical probabilities
- Organize data into lists or charts of equally likely outcomes as a strategy for finding theoretical probabilities (other strategies, such as tree diagrams and the area model, will be introduced in the grade probability unit, What Do You Expect?)
- Use graphs and tallies to summarize and display data
- Use data displayed in graphs and tallies to find experimental probabilities
The overall goal of Connected Mathematics is to help students develop sound mathematical habits. Through their work in this and other data units, students learn important questions to ask themselves about any situation that can be represented and modeled mathematically, such as: What makes an event uncertain? How can we get useful information about such uncertain events? What do we mean by predictable”? When can a series of uncertain events become pre- dictable? Why is probability a mathematics topic? Hou’ can we use mathematics to identify how probable an event may be? What probability is associated with an event that is certain? With an event that could never happen? What kinds of experiments can be performed to find the probability of an event? Can we find a way to compare a probability found mathematically with a probability found experimentally?
The Mathematics in How Likely is it?
The terms chance and probability are applied to situations that have uncertain outcomes on individual trials but a regular pattern of outcomes over many trials. For example, when we toss a coin, we are uncertain whether it will come up heads or tails, but we do know that over the long run, if it is a fair coin, we will get about half heads and half tails. This does not mean we won’t get several heads in a row or that if we get heads now we are more likely to get tails on the next toss. This is a difficult concept for students to grasp: uncertainty on an individual outcome, but predictable regularity in the long run. It often takes a significant amount of time and a variety of experiences that challenge prior conceptions before students understand this basic concept of probability.
If we toss a tack into the air, we know that it will land on either its head or its side. However, if we toss the tack many times, we can use the ratio of the number of times it lands on its side to the total number of tosses to estimate the likelihood that the tack will land on its side. Since we find this ratio through experimentation, it is called an experimental probability
Many uses of probability in daily life are based on experimental probabilities. We collect data for a large number of trials and observe the frequency of a particular result. This is the relative- frequency interpretation of probability. The probability that it will rain or that Shaquille O’Neal will make a free throw are two examples of experimental probabilities based on relative frequencies.
The experimental probability that a coin will land heads up can he expressed as:
We can also determine the theoretical probability of a fair coin landing heads up or tails up by analyzing the situation. If we toss a fair coin, we know that it will land either heads up or tails up and that each outcome is equally likely. Since there are two possible equally likely outcomes, the probability of a fair coin landing heads up is 1 out of 2, or ½. We can write this as P(head) = ½ In general, the theoretical probability that a coin will land heads up can be expressed as:
Another example of a theoretical probability that occurs in this unit involves rolling a number cube. When a number cube is rolled, there are six possible outcomes: 1, 2, 3, 4,5, and 6. Each outcome is equally likely on any roll of the number cube. Thus, P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6. We can use this theoretical lprohahility to make an estimate: if a number cube is tossed many times. we expect each number to occur about 1/6 of the time. We can also compute the probability of events that are made up of more than one equally likely outcome. For example, the theoretical probability of rolling a multiple of 3 on a number cube is 2/6, since two of the six possible equally likely outcomes, 3 and 6, are multiples of 3.
In some situations it is easier to find theoretical probabilities, and in some it is easier to find experimental probabilities. For example, in this unit students will find experimental probabilities of a marshmallow landing on an end or a side when it is tossed, but they will not he able to determine the theoretical probabilities (because although end” and side” are the possible out comes, they are not necessarily equally likely).
Probabilities are useful for predicting what will happen over the long run, yet a theoretical or experimental probability does not tell us exactly what will happen. For example, if we toss a coin 40 times, we may not get exactly 20 heads; but if we toss a coin 1000 times, the fraction of heads will be fairly close to -i-. Experimental data gathered over many trials should produce probabili- ties that are close to the theoretical probabilities (this idea is sometimes called the Law of Large Numbers). If we can calculate a theoretical probability, we can use it to predict what will happen in the long run rather than having to rely on experimentation.
Once we have a probability—theoretical or experimental—we can use it to make predictions. For example, if a coin is tossed 1000 times, we would predict that a head will occur about 500 times. If a number cube is rolled 1000 times, we would predict that a 3 will occur about 1/6 of the time or about 167 times.
It is important for students to realize that a small amount of data may produce wide variation among samples and that only over many trials can we make good estimates for what will happen in the long run. In other words, for our experimental probabilities to be good estimates of the theoretical probabilities, we must base our experimental probabilities on many trials.