# 4.3 Binomial distribution  (Page 2/29)

 Page 2 / 29

A fair coin is flipped 15 times. Each flip is independent. What is the probability of getting more than ten heads? Let X = the number of heads in 15 flips of the fair coin. X takes on the values 0, 1, 2, 3, ..., 15. Since the coin is fair, p = 0.5 and q = 0.5. The number of trials is n = 15. State the probability question mathematically.

P ( x >10)

## Try it

A fair, six-sided die is rolled ten times. Each roll is independent. You want to find the probability of rolling a one more than three times. State the probability question mathematically.

P ( x >3)

Approximately 70% of statistics students do their homework in time for it to be collected and graded. Each student does homework independently. In a statistics class of 50 students, what is the probability that at least 40 will do their homework on time? Students are selected randomly.

a. This is a binomial problem because there is only a success or a __________, there are a fixed number of trials, and the probability of a success is 0.70 for each trial.

a. failure

b. If we are interested in the number of students who do their homework on time, then how do we define X ?

b. X = the number of statistics students who do their homework on time

c. What values does x take on?

c. 0, 1, 2, …, 50

d. What is a "failure," in words?

d. Failure is defined as a student who does not complete his or her homework on time.

The probability of a success is p = 0.70. The number of trials is n = 50.

e. If p + q = 1, then what is q ?

e. q = 0.30

f. The words "at least" translate as what kind of inequality for the probability question P ( x ____ 40).

f. greater than or equal to (≥)
The probability question is P ( x ≥ 40).

## Try it

Sixty-five percent of people pass the state driver’s exam on the first try. A group of 50 individuals who have taken the driver’s exam is randomly selected. Give two reasons why this is a binomial problem.

This is a binomial problem because there is only a success or a failure, and there are a definite number of trials. The probability of a success stays the same for each trial.

## Notation for the binomial: B = binomial probability distribution function

X ~ B ( n , p )

Read this as " X is a random variable with a binomial distribution." The parameters are n and p ; n = number of trials, p = probability of a success on each trial.

It has been stated that about 41% of adult workers have a high school diploma but do not pursue any further education. If 20 adult workers are randomly selected, find the probability that at most 12 of them have a high school diploma but do not pursue any further education. How many adult workers do you expect to have a high school diploma but do not pursue any further education?

Let X = the number of workers who have a high school diploma but do not pursue any further education.

X takes on the values 0, 1, 2, ..., 20 where n = 20, p = 0.41, and q = 1 – 0.41 = 0.59. X ~ B (20, 0.41)

Find P ( x ≤ 12). P ( x ≤ 12) = 0.9738. (calculator or computer)

Go into 2 nd DISTR. The syntax for the instructions are as follows:

To calculate ( x = value): binompdf( n , p , number) if "number" is left out, the result is the binomial probability table.
To calculate P ( x ≤ value): binomcdf( n , p , number) if "number" is left out, the result is the cumulative binomial probability table.
For this problem: After you are in 2 nd DISTR, arrow down to binomcdf. Press ENTER. Enter 20,0.41,12). The result is P ( x ≤ 12) = 0.9738.

## Note

If you want to find P ( x = 12), use the pdf (binompdf). If you want to find P ( x >12), use 1 - binomcdf(20,0.41,12).

The probability that at most 12 workers have a high school diploma but do not pursue any further education is 0.9738.

The graph of X ~ B (20, 0.41) is as follows:

The y -axis contains the probability of x , where X = the number of workers who have only a high school diploma.

The number of adult workers that you expect to have a high school diploma but not pursue any further education is the mean, μ = np = (20)(0.41) = 8.2.

The formula for the variance is σ 2 = npq . The standard deviation is σ = $\sqrt{npq}$ .
σ = $\sqrt{\left(20\right)\left(0.41\right)\left(0.59\right)}$ = 2.20.

what is statistics
statistics is the collection and interpretation of data
Enhance
the science of summarization and description of numerical facts
Enhance
Is the estimation of probability
Zaini
mr. zaini..can u tell me more clearly how to calculated pair t test
Haai
do you have MG Akarwal Statistics' book Zaini?
Enhance
Haai how r u?
Enhance
maybe .... mathematics is the science of simplification and statistics is the interpretation of such values and its implications.
Miguel
can we discuss about pair test
Haai
what is outlier?
outlier is an observation point that is distant from other observations.
Gidigah
what is its effect on mode?
Usama
Outlier  have little effect on the mode of a given set of data.
Gidigah
How can you identify a possible outlier(s) in a data set.
Daniel
The best visualisation method to identify the outlier is box and wisker method or boxplot diagram. The points which are located outside the max edge of wisker(both side) are considered as outlier.
Akash
@Daniel Adunkwah - Usually you can identify an outlier visually. They lie outside the observed pattern of the other data points, thus they're called outliers.
Ron
what is completeness?
I don't get the example
ways of collecting data at least 10 and explain
Example of discrete variable
Gbenga
I am new here, can I get someone to guide up?
alayo
dies outcome is 1, 2, 3, 4, 5, 6 nothing come outside of it. it is an example of discrete variable
jainesh
continue variable is any value value between 0 to 1 it could be 4digit values eg 0.1, 0.21, 0.13, 0.623, 0.32
jainesh
hi
Kachalla
what's up here ... am new here
Kachalla
sorry question a bit unclear...do you mean how do you analyze quantitative data? If yes, it depends on the specific question(s) you set in the beginning as well as on the data you collected. So the method of data analysis will be dependent on the data collecter and questions asked.
Bheka
how to solve for degree of freedom
saliou
Quantitative data is the data in numeric form. For eg: Income of persons asked is 10,000. This data is quantitative data on the other hand data collected for either make or female is qualitative data.
Rohan
*male
Rohan
Degree of freedom is the unconditionality. For example if you have total number of observations n, and you have to calculate variance, obviously you will need mean for that. Here mean is a condition, without which you cannot calculate variance. Therefore degree of freedom for variance will be n-1.
Rohan
data that is best presented in categories like haircolor, food taste (good, bad, fair, terrible) constitutes qualitative data
Bheka
vegetation types (grasslands, forests etc) qualitative data
Bheka
I don't understand how you solved it can you teach me
solve what?
Ambo
mean
Vanarith
What is the end points of a confidence interval called?
lower and upper endpoints
Bheka
Class members write down the average time (in hours, to the nearest half-hour) they sleep per night.
how we make a classes of this(170.3,173.9,171.3,182.3,177.3,178.3,174.175.3)
Sarbaz
6.5
phoenix
11
Shakir
7.5
Ron
why is always lower class bundry used
Caleb
Assume you are in a class where quizzes are 20% of your grade, homework is 20%, exam _1 is 15%,exam _2 is 15%, and the final exam is 20%.Suppose you are in the fifth week and you just found out that you scored a 58/63 on the fist exam. You also know that you received 6/9,8/10,9/9 on the first
quizzes as well as a 9/11,10/10,and 4.5/7 on the first three homework assignment. what is your current grade in the course?
Diamatu
Abdul
if putting y=3x examine that correlation coefficient between x and y=3x is 1.
what is permutation
how to construct a histogram
You have to plot the class midpoint and the frequency
Wydny
ok so you use those two to draw the histogram right.
Amford
yes
Wydny
ok can i be a friend so you can be teaching me small small
Amford
how do you calculate cost effectiveness?
George
Hi everyone, this is a very good statistical group and am glad to be part of it. I'm just not sure how did I end up here cos this discussion just popes on my screen so if I wanna ask something in the future, how will I find you?
Bheka
To make a histogram, follow these steps: On the vertical axis, place frequencies. Label this axis "Frequency". On the horizontal axis, place the lower value of each interval. ... Draw a bar extending from the lower value of each interval to the lower value of the next interval.
Divya
I really appreciate that
I want to test linear regression data such as maintenance fees vs house size. Can I use R square, F test to test the relationship? Is the good condition of R square greater than 0.5
yes of course must have use f test and also use t test individually multple coefficients
rishi
Alright
umar
hi frnd I'm akeem by name, I wanna study economics and statistics wat ar d thing I must do to b a great economist
akeem
Is R square cannot analysis linear regression of X vs Y relationship?
Mok
To be an economist you have to be professional in maths
umar
hi frnds
Shehu