Download 8.1 Introduction to the Binomial Theorem

8.1 Introduction to the Binomial Theorem
Binomial Setting
1) Each observation fits into one of two categories; success or failure.
2) There is a fixed number n of observations.
independent .
3) The n observations are all 4) The probability for success, call it p, is the same for each observation.
If the data is produced in a binomial setting, then the random variable, X = number of successes, is a binomial random variable and the distribution of X is a binomial distribution.
In a random sample of size n from a population with a proportion of p successes, if the population is much larger than the sample, the count X of successes is approximately binomial.
If a distribution is binomial we represent it as B(
n, p)
Is it binomial?
Ex.1) A child's blood type is inherited from his/her parents. If both parents carry the genes for A and O blood, the probability that a child will have type O blood is 0.25. Let X = number of children with type O blood in a family with five children.
Ex. 2) Deal 10 cards from a shuffled deck. Let X be the number of red cards.
Ex. 3) An engineer is inspecting the number of bad light switches that he receives from a company. He finds 10% of the switches are bad.
Ex. 4) Engineers define reliability as the probability that an item will preform its function under specific conditions. If an aircraft engine turbine has a probability of 0.999 of performing properly, let X = number of engines running properly in a fleet of 350 airplanes. 8.1 Formulas for a Binomial Distribution
Binomial Coefficient ( )
Binomial Probability: P(X = k) =
Example) n
k
n!
= k!(n ­ k)!
()
n
k k p (1 ­ p)
The number X of switches that fail inspection has a binomial distribution with n =10 and p = 0.1.
a) Find the probability of exactly 2 switches failing.
b) Find the probability of 0 switches failing.
c) Find the probability that at least one switch fails.
n ­ k