 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Session 8, Part B:
Mathematical Probability (50 minutes)

In This Part: Predicting Outcomes | Fair or Unfair? | Outcomes | Finding the Winner
Making a Probability Table

You can use statistics to determine whether it's possible for a player to develop skill in playing Push Penny. One effective way to analyze the data is to use the principles of probability. Note 3

We cannot know the outcome of a single random event in advance. However, if we repeat the random experiment over and over and summarize the results, a pattern of outcomes begins to emerge. We can determine this pattern by repeating the experiment many, many times, or we can also use mathematical probabilities to describe the pattern. In statistics, we use mathematical probabilities to predict the expected frequencies of outcomes from repeated trials of random experiments.

For example, if you toss a coin, your outcome could either be heads or tails. Since there are two possible outcomes, tossing a fair coin a large number of times would ultimately generate heads for half (or 50%) of the outcomes and tails for half of the outcomes.

When rolling dice, on the other hand, there are six possible outcomes for each die. So if you roll a fair die a large number of times, you would expect a three for about one-sixth of the outcomes, a five for one-sixth of the outcomes, and so forth.

We can use probability tables to express mathematical probabilities. This is the probability table for a fair coin:  Face Frequency Probability  Heads 1 1/2 Tails 1 1/2 This is the probability table for a fair die:  Face Frequency Probability  1 1 1/6 2 1 1/6 3 1 1/6 4 1 1/6 5 1 1/6 6 1 1/6   Problem B1 This spinner uses the numbers one through five, and all five regions are the same size. Create the probability table for this spinner:  Problem B2 Suppose you toss a fair coin three times, and the coin comes up as heads all three times. What is the probability that the fourth toss will be tails? Problem B2 illustrates what is sometimes known as the "gambler's fallacy." Using probability tables, we can predict the outcomes of a toss of one coin or one die. But what if there is more than one coin, or a pair of dice? Problem B3 Toss a coin twice, and record the two outcomes in order (for example, "HT" would mean that the first coin came up heads, and the second coin came up tails).

 a. List all the possible outcomes for tossing a coin twice. How many are there? What is the probability that each occurs?  Note that "HT" is a different outcome than "TH," so both should be listed.   Close Tip Note that "HT" is a different outcome than "TH," so both should be listed.

 b. List all the possible outcomes for tossing a coin three times. How many are there? What is the probability that each occurs?  One way to construct this list is to create two copies of the complete list for two outcomes, and then add H to the end of the first list and T to the end of the second list.    Close Tip One way to construct this list is to create two copies of the complete list for two outcomes, and then add H to the end of the first list and T to the end of the second list.   Session 8: Index | Notes | Solutions | Video