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 Pascalâ€™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.

- Know and apply organized counting techniques such as the
Fundamental Counting Principle.
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.

- Simplify the factorial form of the permutation and combination formula to aid computation.
- Distinguish between situations where replacement is not permitted and situations that permit replacement, including those involving circular arrangements.
- Distinguish between situations where order matters and situations where it does not; select and apply appropriate means of computing the number of possible arrangements of the items in each case.
- Use permutations and combinations to determine the number of possible outcomes of an event.
- Use factorials to define combinations and permutations and to determine the number of ways an event can occur.

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 _{9}*P*_{3}; determine the number of distinct committees of five people that can be formed from a class of 26 students _{26}*C*_{5}.

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?

- Construct and use decision trees.
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.

- Use the concepts of conditional probability to calculate simple probabilities.
- Represent and analyze probabilities of independent events (e.g., repeated tossing of a coin, or throwing dice).

- Recognize and use relative frequency as an estimate for probability.
- If an action is repeated
*n*times and a certain event occurs*b*times, the ratio is called the relative frequency of the event occurring.

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?

- Explain how the Law of Large Numbers defines the relationship between experimental and theoretical probabilities.
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.

- Use simulations to estimate probabilities.
- Use theoretical probability to determine the most likely result if an experiment is repeated a large number of times.

Example: Compare and contrast the binomial distribution (discrete) and the normal distribution (continuous).

- Understand common examples that fit the normal distribution (e.g., height, weight) and examples that do not (salaries, housing prices, size of cities) and explain distinguishing characteristics of each.
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.

- Describe key characteristics (e.g., shape, symmetry/skewness, typical value, and/or spread) of a frequency distribution.
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.

- Use a probability distribution to assess the likelihood of the occurrence of an event.