The Binomial Probability Distribution 6.2 The Binomial Probability Distribution

Binomial Experiment A binomial experiment has the following structure The first test is performed … the result is either a success or a failure The second test is performed … the result is either a success or a failure. This result is independent of the first and the chance of success is the same A third test is performed … the result is either a success or a failure. The result is independent of the first two and the chance of success is the same

Binomial Experiment A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit The card is then put back into the deck A second card is drawn from the deck with the same definition of success. The second card is put back into the deck We continue for 10 cards

Binomial Experiment A binomial experiment is an experiment with the following characteristics The experiment is performed a fixed number of times, each time called a trial The trials are independent Each trial has two possible outcomes, usually called a success and a failure The probability of success is the same for every trial

Notation for Binomial Experiments Notation used for binomial distributions The number of trials is represented by n The probability of a success is represented by p The total number of successes in n trials is represented by X Because there cannot be a negative number of successes, and because there cannot be more than n successes (out of n attempts) 0 ≤ X ≤ n

In our card drawing example (drawing a heart) Each trial is the experiment of drawing one card The experiment is performed 10 times, so n = 10 The trials are independent because the drawn card is put back into the deck Each trial has two possible outcomes, a “success” of drawing a heart and a “failure” of drawing anything else The probability of success is 0.25, the same for every trial, so p = 0.25 X, the number of successes, is between 0 and 10

Success? The word “success” does not mean that this is a good outcome or that we want this to be the outcome A “success” in our card drawing experiment is to draw a heart If we are counting hearts, then this is the outcome that we are measuring There is no good or bad meaning to “success”

Binomial Probability Formula The general formula for the binomial probabilities is just this For P(x), the probability of x successes, the probability is The number of ways of choosing x out of n, times The probability of x successes, times The probability of n-x failures This formula is P(x) = nCx px (1 – p)n-x

Calculators We do not need to use the formula, we will be using the calculator commands instead. We will do more of these at the end using technology

Mean of a Binomial Distribution We would like to find the mean (expected value) of a binomial distribution. Example There are 10 MC questions on a quiz The probability of success is .20 on each one Then the expected number of successes (correct guesses on the quiz) would be 10 • .20 = 2 The general formula: μX = n p

Variance of a Binomial Distribution We would like to find the standard deviation and variance of a binomial distribution This calculation is more difficult The standard deviation is σX = 𝒏 𝒑 (𝟏 – 𝒑) and the variance is σX2 = n p (1 – p)

Example For our random guessing on a quiz problem Therefore The mean is np = 10 • .2 = 2 The variance is np(1-p) = 10 • .2 • .8 = 1.6 The standard deviation is √1.6 = 1.26 Remember the empirical rule? A passing grade of 6 is 10 standard deviations from the mean …

Binomial Probability Distribution With the formula for the binomial probabilities P(x), we can construct histograms for the binomial distribution There are three different shapes for these histograms When p < .5, the histogram is skewed right When p = .5, the histogram is symmetric When p > .5, the histogram is skewed left

Skewed Right For n = 10 and p = .2 (skewed right) Mean = 2 Standard deviation = .4

Skewed Left For n = 10 and p = .8 (skewed left) Mean = 8 Standard deviation = .4

Symmetric For n = 10 and p = .5 (symmetric) Mean = 5 Standard deviation = .5

Shape Despite binomial distributions being skewed, the histograms appear more and more bell shaped as n gets larger This will be important!

Calculators BinomCdf and BinomPdf

Calculator Instructions MENU 5:Probability, 5:Distributions, D: BinomialPDF Use PDF if you are finding probability of a specific number of successes. Enter n(# of trials), p(probability of success), x (# of successes desired) If you leave x blank, you will get a list of all possible successes not just a certain # MENU 5:Probability, 5:Distributions, E: BinomialCDF Use CDF if you are finding the probability of a range of successes. Enter n(# of trials), p(probability of success), then the lower and upper bounds of the range you are finding

Example Suppose 70% of all Americans have cable TV. P(x) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Suppose 70% of all Americans have cable TV. In a Random sample of 15 households, Construct a binomial probability distribution.

Example Suppose 70% of all Americans have cable TV. In a Random sample of 15 households, what is the probability that exactly 10 households have cable? Binompdf(n, p, x) Binompdf(15, .7, 10) Use pdf when exactly is used!!! Use Calculator and: P(x = 10) =

Another Example Part 2 Suppose 70% of all Americans have cable TV. In a Random sample of 15 households, what is the probability that at least 13 households have cable? P( x 13) = P(13) or P(14) or P(15)

Another Example Part 3 Suppose 70% of all Americans have cable TV. In a Random sample of 15 households, what is the probability that fewer than 13 households have cable? P(x 12)= P(1)+P(2)….P(12) Notice the 12, NOT 13!!

Example (#35) According to flightstats.com, American Airlines flight 1247 from Orlando to Los Angeles is on time 65% of the time. Suppose fifteen flights are randomly selected, and the number of on-time flights is recorded. Explain why this is a binomial experiment Find the probability that exactly 10 flights are on time. Find the probability that at least 10 flights are on time. Find the probability that fewer that 10 flights are on time. Find the probability that between 7 and 10 flights, inclusive, are on time.

Example (#49) According to the Uniform Crime Report, 2006, nationwide, 61% of murders committed in 2006 were cleared by arrest or exceptional means. For 250 randomly selected murders committed in 2006, compute the mean and standard deviation of the random variable x, the number of murders cleared by arrest or exceptional means. Interpret the mean Of the 250 randomly selected murders, find the interval that would be considered “unusual” for the number of murders that was cleared.