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This unit focuses on constructing and interpreting binomial distributions as part of data analysis and probability. Students will learn how to create probability histograms for discrete random variables using both experimental and theoretical probabilities according to Georgia Performance Standards MM3D1. Key concepts include random and discrete random variables, probability distributions, and the characteristics of binomial experiments. Through various examples, including quizzes and surveys, students will explore calculations of success probabilities and analyze the shapes of distributions, distinguishing between symmetric and skewed distributions.
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Unit 6 – Data Analysis and Probability 6.1 – Construct and interpret binomial distributions
Georgia Performance Standards MM3D1 – Students will create probability histograms of discrete random variables, using both experimental and theoretical probabilities.
Vocabulary A random variable is a variable whose value is determined by the outcomes of a random event. A discrete random variable is a variable that can take on only a countable number of distinct values. A continuous random variable is a variable that can take on an uncountable, infinite number of possible values, often over a specified interval. A probability distribution is a function that gives the probability of each possible value of random variable. The sum of all the probabilities in a probability distribution must equal 1.
Vocabulary A binomial distribution shows the probabilities of the outcomes of a binomial experiment. A binomial experiment has n independent trials, with two possible outcomes (success or failure) for each trial. The probability for success is the same for each trial. The probability of exactly k success in n trials is P(k successes) = nCkpk(1-p)n-k Formula for success
Vocabulary A probability is symmetric if a vertical line can be drawn to divide the histogram of the distribution into two parts that are mirror images. A distribution that is not symmetric is called skewed.
Binomial Distribution Probability Distribution - symmetric Skewed Probability Distribution
Construct a Probability Distribution (Ex 1) Let X be a random variable that represents the number of questions that students guessed correctly on a quiz with three true-false questions. Make a table and a histogram showing the probability distribution for X.
Interpret a Probability Distribution (Ex 2) Use the probability distribution in Example 1 to find the probability that a student guesses at least two questions correctly. What are some things to consider??
Construct a binomial distribution (Ex 3) In a standard deck of cards, 25% are hearts. Suppose you choose a card at random, note whether it is a heart, then replace it. You conduct the experiment 5 times. Draw a histogram of the binomial distribution for your experiment. P(k successes) = nCkpk(1-p)n-k
Construct a binomial distribution (Ex 3) In a standard deck of cards, 25% are hearts. Suppose you choose a card at random, note whether it is a heart, then replace it. You conduct the experiment 5 times. Draw a histogram of the binomial distribution for your experiment. P(k=0)= 5C0(0.25)0(0.75)5 = 0.237 P(k=1)= 5C1(0.25)1(0.75)4 =0.396 P(k=2)= 5C2(0.25)2(0.75)3 = 0.264 P(k=3)= 5C3(0.25)3(0.75)2 =0.088 P(k=4)= 5C4(0.25)4(0.75)1 =0.016 P(k=5)= 5C5(0.25)5(0.75)0 =0.001
Construct a binomial distribution (Ex 4) According to a recent survey (It must be true if it’s in a PowerPoint), about 85% of the population at CHS love the show The Walking Dead. Suppose you ask 4 random students if they like the show The Walking Dead. Draw a histogram of the binomial distribution showing the probability that exactly k of the students like the show.
What can we learn from binomial distributions? What is the least likely outcome from the survey? What is the probability that k = 1? Describe the shape of the binomial distribution?
