Successful students will be able to quantify the likelihood that an event will occur through combinatorics and other counting principles, relative frequency, distributions, and the comparison of theoretical probability to simulations. Also included are binomial expansion and the relationship to Pascal’s triangle and binomial distributions. There are a variety of types of test items including some that cut across the objectives in this standard and require students to make connections and solve rich contextual problems.
The Fundamental Counting Principle is a way of determining the number of ways a sequence of events can take place. If there are n ways of choosing one thing and m ways of choosing a second after the first has been chosen, then the Fundamental Counting Principle says that the total number of choice patterns is n x m.
Example: Interpret 6! as the product 6 · 5 · 4 · 3 · 2 · 1; recognize that = 15 · 14 · 13 = 2730.
Example: Determine the number of different license plates that can be formed with two letters and three numerals; determine how many are possible if all the letters have to come first; determine how many different subsets are possible for a set of six elements?
Example: Determine the number of different four-digit numbers that can be formed if the first digit must be non-zero and each digit may be used only once; determine how many are possible if the first digit must be non-zero but digits can be used any number of times.
Example: Determine the number of ways that first, second, and third place could be awarded in a race having 9 contestants if there are no ties 9P3; determine the number of distinct committees of five people that can be formed from a class of 26 students 26C5.
Task related to this benchmark: How Odd . . .
Example: A basketball player has an average success rate of 60% on free throws. What is the probability that she will make 10 out of 12 free throws in the next game?
Task related to this benchmark: Is Your Score Normal?
A tree is a connected graph containing no closed loops (cycles).
Note: Tree diagrams can also be used to analyze games such as tic-tac-toe or Nim or to simply organize outcomes.
Example: In a sample of 100 randomly selected students, 37 of them could identify the difference in two brands of soft drinks. Based on these data, what is the best estimate of how many of the 2352 students in the school could distinguish between the soft drinks?
The law of large numbers indicates that if an event of probability p is observed repeatedly during independent repetitions, the ratio of the observed frequency of that event to the total number of repetitions approaches p as the number of repetitions becomes arbitrarily large.
Example: Compare and contrast the binomial distribution (discrete) and the normal distribution (continuous).
The normal (or Gaussian) distribution is actually a family of mathematical distributions that are symmetric in shape with scores more concentrated in the middle than in the tails. They are sometimes described as bell shaped. Normal distributions may have differing centers (means) and scale (standard deviation). The standard normal distribution is the normal distribution with a mean of zero and a standard deviation of one. In this distribution, approximately 68% of the data lie within one standard deviation of the mean and 95% within two.
A frequency distribution shows the number of observations falling into each of several ranges of values; if the percentage of observations is shown, the distribution is called a relative frequency distribution. Both frequency and relative frequency distributions are portrayed through tables, histograms, or broken-line graphs.