Question 1206968
<font color=black size=3>
Answer:  <font color=red size=4>0.8392</font>


Explanation


Binomial formula
B(x) = (nCx)*(p^x)*(1-p)^(n-x)
where,
n = sample size = 14
p = probability a person prefers a brown car = 0.30
x = number of people who prefer a brown car
x spans from x = 0 to x = 14.
i.e. x is from the set {0,1,2,..,13,14}


The nCx refers to the nCr combination formula. 
Some textbooks call it the "choose" formula. 


Let's compute the binomial probability that x = 0 people prefer a brown car.
B(x) = (nCx)*(p^x)*(1-p)^(n-x)
B(x) = (14Cx)*(0.3^x)*(1-0.3)^(14-x)
B(0) = (14C0)*(0.3^0)*(0.7)^(14-0)
B(0) = (1)*(0.3^0)*(0.7)^(14-0)
B(0) = 0.00678223 which is approximate
There's about a 0.678% chance of 0 people in the sample preferring a brown car (i.e. that everyone in the sample prefers some other color than brown).


Follow similar steps for x = 1 and x = 2 to find:
B(1) = 0.04069338
B(2) = 0.11336014
These values are approximate.


Then,
B(0)+B(1)+B(2) = 0.00678223 + 0.04069338 + 0.11336014 = 0.16083575
There's about a 16.08% chance of selecting between x = 0 and x = 2 people who prefer a brown car.


Subtract this from 1 to get the final answer. This is because:
<font color=blue>B(0)+B(1)+B(2)</font> + <font color=red>B(3)+B(4)+B(5)+...+B(13)+B(14)</font> = 1
which leads to
<font color=red>B(3)+B(4)+B(5)+...+B(13)+B(14)</font>  = 1- ( <font color=blue>B(0)+B(1)+B(2)</font> )
The stuff in red is the answer we want while the stuff in blue is what we calculated earlier.


So,
1 - 0.16083575 = 0.83916425
This rounds to <font color=red>0.8392</font> when rounding to the nearest ten-thousandth aka rounding to 4 decimal places.


--------------------------------------------------------------------------------------------------------------------------


As the other tutor mentions, there are fast shortcuts using technology.


TI83 or TI84:
<font color=red>1 - binomCDF(14, 0.3, 2)</font>
The binomCDF function is found by pressing the button labeled "2nd" and then the VARS key. 
Scroll down a bit until you find the function. 


Spreadsheet:
<font color=red>=1-BinomDist(2,14,0.3,1)</font>
Don't forget about the equal sign up front. If that isn't typed in, then the spreadsheet will think you are entering plain text, and the formula will not evaluate. 
The "1" at the end of the command tells the spreadsheet to use a CDF rather than a PDF.
Refer to the help manual for more information.


GeoGebra:
Open up the probability calculator viewport and select "binomial" under the list of distributions. 
Type in the parameters n = 14, p = 0.3 and you should see a table of values show up. 
Select values x = 3 through x = 14 to have it add up the proper B(x) values we want.
Or you can type 3 and 14 into the input boxes that correspond to P(__ < x < ___) 


There are many online calculators that are great alternatives.
Feel free to explore your favorite.
</font>