Binomial Probability Section 8.1.1

Starter Here’s a game you will like: Let’s bet a dollar on this proposition: I will roll a fair die once. If it comes up 1 or 2, I win. If it comes up 3, 4, 5, or 6, you win! –What is the probability that I win? –What is the probability that I lose? –How much should we each bet to make this a fair game?

Objectives Identify whether a random variable is in a binomial setting Use a calculator command to find the PDF of a binomial random variable Express the PDF as either a table or a histogram California Standard 7.0 Students demonstrate an understanding of the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to those families.

Two-Outcome Probability Questions In many probability questions, there are only two possible outcomes. –Shoot a free throw and make or miss the shot –Play roulette and bet red or black –Play today’s starter dice game and win or lose In such cases, the probabilities of winning or losing (also called success and failure) are easy to compute on the calculator or by a fairly simple formula –Today we will use the calculator, tomorrow the formula This easy calculator approach can be used if the problem meets 4 conditions, called the binomial setting.

The Binomial Setting 1.There are only two outcomes: success and failure 2.There is a pre-determined number of trials You know how many times you will do the activity before doing it the first time 3.Each trial is independent of all other trials 4.The probability of success is the same for each trial

Example: The Starter Continued Does the starter problem meet the 4 conditions needed to call it binomial? –Two outcomes Yes: Rolls of 1 or 2 are success, 3 – 6 are failure –Fixed number of trials Yes if we agree in advance how many rolls to do –Independent trials Yes: Each roll of the die is unaffected by others –Fixed probability Yes: P(success) = 1/3 for each roll

Finding the Binomial PDF In binomial probability, the variable X is the number of successes achieved in n trials. If you perform n trials, what is the least number of successes you could have? –You could have zero successes What is the greatest number of successes? –You could have n successes So X can take on integers from 0 to n –Now how do we find the probability of each?

Finding the Binomial PDF The calculator has a binomal PDF command in the DISTR menu –Tap 2 nd :DISTR and scroll down to binompdf( The syntax is to first enter n (the number of trials you will do), then p (the fixed probability of success), and then X (the number of successes you are interested in. –Example: binompdf(5, 1/3, 3) would give the probability of exactly 3 wins of the starter dice game in 5 trials where p(success) = 1/3 –Try it now. You should get P(X=3) =.1646

Example: The Starter Continued Imagine that we agree to play the dice game 5 times. –What are the possible values for X? Use your calculator to find the probability of each possible value of X (round to.01) –Write the PDF of X X P(X).16

Example: The Starter Continued Imagine that we agree to play the dice game 5 times. There are 6 possible values for X. What are they? Use your calculator to find the probability of each possible value of X (round to.01) Write the PDF of X X P(X).13.16

Example: The Starter Continued Imagine that we agree to play the dice game 5 times. There are 6 possible values for X. What are they? Use your calculator to find the probability of each possible value of X (round to.01) Write the PDF of X X P(X)

Displaying the PDF as a Histogram You can now enter the X values in L 1 and the P(X) values in L 2 and get the calculator to display the PDF histogram. Put the X values in L 1 manually or by the sequence command Put the probabilities in L 2 by this simple command: binompdf(5,1/3)→L 2 –Note that omitting the X value causes the calculator to generate ALL probabilities at once! Set up STAT PLOT 1 to be a histogram based on L 1 and L 2. Think about the window you want. Sketch the resulting graph on paper and write a brief verbal description of the distribution. –Note that there is a similar example on page 420 that shows all the calculator commands.

A New Example Suppose that Chris is a basketball player who normally makes 75% of his free throws. Assume that he takes 6 free throw shots tonight. Let X be the number of shots he makes. –Is this a binomial setting? Check 4 conditions –Create the PDF in list form on your calculator –Create the PDF histogram and sketch it in your notes. –Comment on a significant difference between this distribution and the one we found for the dice game. What do you think caused this difference? The dice problem was right-skewed because p was less than one half. The free throw problem was left-skewed because p was greater than one half.

Objectives Identify whether a random variable is in a binomial setting Use a calculator command to find the PDF of a binomial random variable Express the PDF as either a table or a histogram California Standard 7.0 Students demonstrate an understanding of the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to those families.

Homework Read pages 416 – 420 Do problems 1 – 4 and 5 (a – d)