1. Unit 6 – Data Analysis and Probability
6.1 – Construct and interpret binomial distributions

2. Georgia Performance Standards
MM3D1 – Students will create probability histograms of discrete random variables, using both experimental and theoretical probabilities.

3. 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.

4. 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

5. 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.

6. Binomial Distribution

7. Binomial Distribution
Probability Distribution - symmetric
Skewed Probability Distribution

8. 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) 1 2 3
Outcomes
P(X)
1/8
3/8

9. Construct a Probability Distribution (Ex 1)
X (number correct) 1 2 3
Outcomes
P(X)
1/8
3/8

10. 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) 1 2 3
Outcomes
P(X)
1/8
3/8

11. 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

12. 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

13. 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.

14. 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?