Download Chapter 14: Binomial Distributions Binomial Probability Distributions

Chapter 14: Binomial Distributions
Binomial Probability Distributions is one of Finite Probability Distribution.
Requirements of Binomial Probability Distributions
1) The procedure has a fixed number of observations (n).
2) The observations must be independent.
3) Each observation must have all outcomes classified into two categories.
(Success or Failure) (Yes or No) (Favorable or Unfavorable)
4) The probability of success (p) remains the same in all trials.
Notation for Binomial Probability Distributions:
S or F (success or failure)
P (S) = p (p = probability of a success. )
P (F ) = 1 − p = q (q = probability of a failure. )
n: the fixed number of trials.
k: the number of successes among n trials (k = 0, 1, 2 . . . , n).
Ex. There are 100 people.
Question: Do you like Math?
They have only two choices: Yes or No.
How many people are there who said ”yes”?
Ex. There are 10 multiple choice questions.
Question: Is my answer correct or not?
There is only one result: correct or incorrect.
How many right answers did I have on the test?
Ex. Within five of tossing a coin.
Question: Is it a head or not?
There is only one result: a head or a tail.
How many heads did I have for the coin tossing?
Ex. Find the probability for each value of X. A coin is tossed three times. Find the probability
of getting exactly two heads. (50%)
Value of X = k
Probability P (X = k)
k
P (k)
—————————————————————————
0
1
2
3
Binomial Probability Formula:
P (X = k) =
n!
· pk · (1 − p)n−k ,
(n − k)! k!
k = 0, 1, 2, . . . , n.
Binomial Coefficient
( )
n
n!
=
k
(n − k)! k!
Ex. A survey from a research found that 30% of teenager have a part-time jobs. If four
teenagers are selected at random find the probabilities P (k) and fill the table.
Value of X = k
Probability P (X = k)
k
P (k)
—————————————————————————
0
1
2
3
4
How to find the probability for each value of X.
1) Binomial Probability Formula:
P (X = k) =
n!
· pk · (1 − p)n−k ,
(n − k)! k!
k = 0, 1, 2, . . . , n.
2) Binomial Probability Table:
3) Using a computer:
Mean, Variance, and Standard Deviation for Binomial Distribution
Mean: µ = n · p
Variance: σ 2 = n · p · q
Standard Deviation: σ =
√
n·p·q
Range Rule of Thumb
Maximum usual value: µ + 2σ
Minimum usual value: µ − 2σ
Ex. Assume that the given procedure yields a binomial distribution with n trials and
the probability of success for one trials is p. Use the given values of n and p to find the µ and
σ. Also,use the range of rule of thumb to find the minimum
usual value and the maximum usual value. n = 20, p = 0.4.
Binomial Probability Distributions
Ex. A survey from a research found that 30% of teenager have a part-time jobs.
If five teenagers are selected at random find the probability that three of them
will have part-time jobs.
Value of X = k
Probability P (X = k)
k
P (k)
—————————————————————————
0
1
2
3
4
5
Ex. Find the probability that at least one of them will have part-time jobs.
At least one ⇒ Find the individual probability for 1, 2, 3, 4, and 5 and add them to get
the total probability.
Normal as Approximation to Binomial :
A binomial distribution is discrete. However, when n increases it is approximated by a normal
distribution that is a continuous distribution.
The binomial distribution with n and p can be approximated by a normal distribution
with mean np and variance npq.
If the following conditions are satisfied:
n is large.
np ≥ 10, and n(1 − p) ≥ 10.
p is not close to 0 or 1.
It is desirable that p is close to 0.5.