Probability : Binomial

오늘의 개념 

"Binomial

In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial; when n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.

The binomial distribution is frequently used to model the number of successes in a sample of sizen drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution is a good approximation, and widely used.

** http://en.wikipedia.org/wiki/Binomial_distribution

Example(1) :Flip 2 coins

Let Y= total # heads

P(Y=0)=1/4
P(Y=1)=1/2  (<- 요것들을 probability distribution 이라고 부릅니다.)
P(Y=2)=1/4


Binomial(2,1/2) : 해석- we give 2 experiments in this case that put the coin twice, and probability or head each time equal to a half (1/2)

Example(1) :Flip 1 coin


Let Y= total # heads

P(Y=0)=1/2
P(Y=1)=1/2


Binomial(1,1/2) => 특별한 경우, 한번의 기회, 확률은 반반. (Bernoulli distribution)

Example(1) :Flip 3 coins (three times)


Let Y= total # heads

P(Y=0)=1/8
P(Y=1)=3/8
P(Y=2)=3/8
P(Y=3)=1/8

Binominal(3,1/2)


** Advanced concept**

Binomial Probability Formula

A probability formula for Bernoulli trials. The probability of achieving exactly k successes in n trials is shown below.

Formula:	n = number of trials k = number of successes n – k = number of failures p = probability of success in one trial q = 1 – p = probability of failure in one trial
Example:	You are taking a 10 question multiple choice test. If each question has four choices and you guess on each question, what is the probability of getting exactly 7 questions correct? n = 10 k = 7 n – k = 3 p = 0.25 = probability of guessing the correct answer on a question q = 0.75 = probability of guessing the wrong answer on a question

source: http://www.mathwords.com/b/binomial_probability_formula.htm