SOLUTION: A researcher wishes to conduct a study of the color preference of new car buyers. Suppose that 30% of this population prefers the color brown. If 14 buyers are randomly selected, w

Question 1206968: A researcher wishes to conduct a study of the color preference of new car buyers. Suppose that 30% of this population prefers the color brown. If 14 buyers are randomly selected, what is the probability that at least 3 buyers would prefer brown? Round your answer to the nearest ten-thousandth, if necessary.
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
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A researcher wishes to conduct a study of the color preference of new car buyers.
Suppose that 30% of this population prefers the color brown.
If 14 buyers are randomly selected, what is the probability that at least 3 buyers
would prefer brown? Round your answer to the nearest ten-thousandth, if necessary.
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It is a Binomial experiment with n= 14 trials and the probability of the individual success p= 0.3. They want you find the probability of the event (n=14; k>= 3; p=0.3). You can use an online calculator at the site https://stattrek.com/online-calculator/binomial and to fet the answer instantly. P = 0.83916. Alternatively, you may use your regular calculator TI-83/84 with irs standard function binomCFD n, p, c <<<---=== formatting pattern P = 1 - binomcfd(14, 0.3, 2). Alternatively, you may use the Excel standard function BINOM.DIST(3,14,0.3,1). In any case, you will get the same answer. The links to read - for binomcfd https://www.mathbootcamps.com/binomial-probabilities-ti-83-or-84-calculator/ - for BINOM.DIST https://support.microsoft.com/en-us/office/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c

Solved.

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If you want to see many similar or different solved problems on binomial distribution probability, look into the lessons
    - Simple and simplest probability problems on Binomial distribution
    - Typical binomial distribution probability problems
    - How to calculate Binomial probabilities with Technology (using MS Excel)
    - Solving problems on Binomial distribution with Technology (using MS Excel)
    - Solving problems on Binomial distribution with Technology (using online solver)
in this site.

After reading these lessons, you will be able to solve such problems on your own,
which is your PRIMARY MAJOR GOAL visiting this forum (I believe).

Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Answer: 0.8392

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:
B(0)+B(1)+B(2) + B(3)+B(4)+B(5)+...+B(13)+B(14) = 1
which leads to
B(3)+B(4)+B(5)+...+B(13)+B(14) = 1- ( B(0)+B(1)+B(2) )
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 0.8392 when rounding to the nearest ten-thousandth aka rounding to 4 decimal places.

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

TI83 or TI84:
1 - binomCDF(14, 0.3, 2)
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:
=1-BinomDist(2,14,0.3,1)
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.

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