Binomial Distributions Special Probability Distributions

Objectives: To determine the characteristic of a special type for probabilities distribution (Binomial Distributions) To apply knowledge of Binomial distributions in several real world settings

Binomial Distributions We have Binomial trials if: there are two possible outcomes (success and failure). These outcomes must be mutually exclusive (disjoint) the probability of success, p, is constant. the trials are independent. There is a fixed number of observations

The Binomial Model A Binomial model tells us the probability for a random variable that counts the number of successes in a fixed number of trials. Two parameters define the Binomial model: n, the number of trials; and, p, the probability of success. We denote this B(n, p).

Examples: Indentifying Binomials Which of the following are binomial experiments? (a) A player rolls a pair of fair die 10 times. The number X of 7’s rolled is recorded. (b) The 11 largest airlines had an on-time percentage of 84.7% in November, 2001 according to the Air Travel Consumer Report. In order to assess reasons for delays, an official with the FAA randomly selects flights until she finds 10 that were not on time. The number of flights X that need to be selected is recorded. (c ) In a class of 30 students, 60% are female. The instructor randomly selects 4 students. The number X of females selected is recorded. Yes No. No Set # of trials. No. p is not constant.

The Binomial Model Binomial probability model for Bernoulli trials: B(n,p) n = number of trials p = probability of success q = 1 – p = probability of failure X = # of successes in n trials P(X=k) = nCk • pk • qn-k

Binomial Coefficient Same formula for Combinations or nCr The number of ways of arranging k successes among n observations is given by the binomial coefficient: For k = 0, 1, 2,…, n Same formula for Combinations or nCr

Review: Types of questions asked for a binomial distribution

Examples P(X=5) = 20C5 • (.183)5 • (.817)15 = 0.1535 According to the United States Census Bureau, 18.3% of all households have 3 or more cars. In a random sample of 20 households, what is the probability that exactly 5 have 3 or more cars? (b) In a random sample of 20 households, what is the probability that less than 4 have 3 or more cars? (c) In a random sample of 20 households, what is the probability that at least 4 have 3 or more cars? P(X=5) = 20C5 • (.183)5 • (.817)15 = 0.1535 P(X<4) = P(X=0) + P(X=1) + P(X=2) + P(X=3)= .4885 P(X≥4) = P(X=4) +…+ P(X=20) = 1 - P(X≤3)=.5115

Probability vs Cumulative pdf: Given a random variable X, the probability distribution function (pdf) assigns a probability to each value of X. The probabilities must satisfy the rules for probabilities given in Chapter 6. cdf: Given a random variable X, the cumulative distribution function (cdf) of X calculates, the sum of the probabilities for 0, 1, 2,…, up to the value X. That is, it calculates the probability of obtaining at most X successes in n trials

Calculator: Back to the Distributions Menu 2ndVARS Binomialpdf(n,p,x) – this calculates the probability of a single binomial P(X = k) Binomialcdf(n,p,x) – this calculates the cumulative probabilities from P(X=0) to P(X=k)

Expected Value & Standard Deviation

Example: Mean & SD According to the United States Census Bureau, 18.3% of all households have 3 or more cars. In a simple random sample of 400 households, determine the mean and standard deviation number of households that will have 3 or more cars.

Independence One of the important requirements for Binomial trials is that the trials be independent. When we don’t have an infinite population, the trials are not independent. But, there is a rule that allows us to pretend we have independent trials: The 10% condition: Binomial trials must be independent. If that assumption is violated, it is still okay to proceed as long as the sample is smaller than 10% of the population. This is because, when using a small percentage of a given population, the binomial model provides similar result to the true distribution (the hypergeometric model for dependent outcomes)

Example: Constructing a Binomial Distribution According to the Air Travel Consumer Report, the 11 largest air carriers had an on-time percentage of 84.7% in November, 2001. Suppose that 4 flights are randomly selected from November, 2001 and the number of on-time flights X is recorded. Construct a probability distribution for the random variable X. X P(X)

Closing Question: What characteristics of a given situation should we look for to determine whether or not a given random variable fits the Binomial Model?