6.1 – CONSTRUCT AND INTERPRET BINOMIAL

DISTRIBUTIONS

Unit 6 – Data Analysis and Probability

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

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. X (number correct)

0 1 2 3

Outcomes 1 3 3 1

P(X) 1/8 3/8 3/8 1/8

Construct a Probability Distribution (Ex 1)

X (number correct)

0 1 2 3

Outcomes 1 3 3 1

P(X) 1/8 3/8 3/8 1/8

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??

X (number correct)

0 1 2 3

Outcomes 1 3 3 1

P(X) 1/8 3/8 3/8 1/8

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?