**Title:** Birthday Paradox

**Description:**
This Illuminations lesson demonstrates the birthday paradox, using it as a springboard into a unit on probability. Students use the TI-83 graphing calculator to run a Monte Carlo simulation with the birthday paradox and engage in a graphical analysis of the birthday-problem function.

**Standard(s): **

[MA2013] (6) 25: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. [6-SP1]

[MA2013] (6) 26: Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. [6-SP2]

[MA2013] (6) 27: Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. [6-SP3]

[MA2013] (6) 29: Summarize numerical data sets in relation to their context, such as by: [6-SP5]

[MA2013] (7) 17: Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. [7-SP1]

[MA2013] (7) 18: Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. [7-SP2]

[MA2013] (7) 21: Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ^{1}/_{2} indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. [7-SP5]

[MA2013] (7) 22: Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. [7-SP6]

[MA2013] (7) 23: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. [7-SP7]

[MA2013] (7) 24: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. [7-SP8]

[MA2013] AL2 (9-12) 46: (+) Use permutations and combinations to compute probabilities of compound events and solve problems. [S-CP9]

[MA2013] GEO (9-12) 42: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [S-MD6]

[MA2013] AL2 (9-12) 33: Write a function that describes a relationship between two quantities.* [F-BF1]

[MA2013] AL2 (9-12) 37: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [S-MD6]

[MA2013] PRE (9-12) 44: Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [S-IC1]

[MA2013] PRE (9-12) 45: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. [S-IC2]

[MA2013] PRE (9-12) 46: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [S-IC3]

[MA2013] PRE (9-12) 47: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. [S-IC4]

[MA2013] ALT (9-12) 41: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [S-MD6]

[MA2013] DM1 (9-12) 12: Use combinatorial reasoning and counting techniques to solve application-based problems. (Alabama)

[MA2013] PRE (9-12) 50: (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. [S-MD1]

[MA2013] PRE (9-12) 51: (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. [S-MD2]