Binomial Probability Finite 8 4 Binomial probability distributions

Binomial Probability Finite 8 -4

• Binomial probability distributions allow us to deal with circumstances in which the outcomes belong to two relevant categories such as acceptable/defective or survived/died. Binomial Distribution

A binomial probability distribution results from a procedure that meets all the following requirements: 1. The procedure has a fixed number of trials. 2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials. ) 3. Each trial must have all outcomes classified into two categories (commonly referred to as success and failure). 4. The probability of a success remains the same in all trials. Binomial Probability Distribution

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. Yes, Binomial (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. No, not set amount (c ) In a class of 30 students, 55% are female. The instructor randomly selects 4 students. The number X of females selected is recorded. No, Probability changes EXAMPLE Identifying Binomial Experiments

S and F (success and failure) denote the two possible categories of all outcomes; p and q will denote the probabilities of S and F, respectively, so P(S) = p (p = probability of success) P(F) = 1 – p = q (q = probability of failure) Notation for Binomial Probability Distributions

n denotes the fixed number of trials. x denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive. p denotes the probability of success in one of n trials. the q denotes the probability of failure in one of the n trials. P(x) denotes the probability of getting exactly x successes among the n trials. Notation (continued)

v Be sure that x and p both refer to the same category being called a success. v When sampling without replacement, consider events to be independent if n < 0. 05 N. Important Hints

n ! P(x) = • px • qn-x (n – x )!x! for x = 0, 1, 2, . . . , n where n = number of trials x = number of successes among n trials p = probability of success in any one trial q = probability of failure in any one trial (q = 1 – p) Using the Binomial Probability Formula

n ! P(x) = • px • (n – x )!x! qn-x The number of outcomes with exactly x successes among n trials Rationale for the Binomial Probability Formula

n ! P(x) = • px • (n – x )!x! Number of outcomes with exactly x successes among n trials qn-x The probability of x successes among n trials for any one particular order Binomial Probability Formula

According to the United States Census Bureau, 18. 3% of all households have 3 or more cars. (a) In a random sample of 20 households, what is the probability that exactly 5 have 3 or more cars? P(X=5) = 20 C 5 • (. 183)5 • (. 817)15 = 0. 1535 EXAMPLE

According to the United States Census Bureau, 18. 3% of all households 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? P(X<4) = P(X=0) + P(X=1) + P(X=2) + P(X=3)=. 4885 EXAMPLE

According to the United States Census Bureau, 18. 3% of all households 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≥ 4) = P(X=4) +…+ P(X=20) = 1 - P(X≤ 3)=. 5115 EXAMPLE

• If a coin is tossed ten times then what will be the probability for getting 7 heads and 3 tails in 10 tosses? Example

• In a store, out of all the people who came there thirty percent bought a shirt. If four people came in the store together then find the probability of one of them buying a shirt. Example

• In a hospital sixty percent of patients are dying of a disease. If on a certain day, eighth patients got admitted in the hospital for that disease what are the chances of three to survive? Example

• Suppose we are throwing a dice thrice. Find the probability of finding a multiple of 3 in one of the throws. Example

• In a restaurant seventy percent of people order for Chinese food and thirty percent for Italian food. A group of three persons enter the restaurant. Find the probability of at least two of them ordering for Italian food. • Probability for two ordering Italian food, Example

• In a restaurant seventy percent of people order for Chinese food and thirty percent for Italian food. A group of three persons enter the restaurant. Find the probability of at least two of them ordering for Italian food. • Probability for all three ordering Italian food Example

• In a restaurant seventy percent of people order for Chinese food and thirty percent for Italian food. A group of three persons enter the restaurant. Find the probability of at least two of them ordering for Italian food. • Hence, the probability for at least two persons ordering Italian food is, Example

• In an exam only ten percent students can qualify. If a group of 4 students have appeared, find the probability that at most one student will qualify? Example

• A basket contains 70 good apples and 30 are spoiled. Three apples are drawn at random from basket. What is the probability that of the 3 apples, Exactly two good apples Example

• A basket contains 70 good apples and 30 are spoiled. Three apples are drawn at random from basket. What is the probability that of the 3 apples, At least one is good Example

• A basket contains 70 good apples and 30 are spoiled. Three apples are drawn at random from basket. What is the probability that of the 3 apples, At most two are good Example