Question 1208472
<font color=black size=3>
Answer:  <font color=red>0.88618</font>  (approximate)


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


<font size=4>Quick way to find the answer using technology</font>


On a TI83 or similar, you would use the <a href="https://www.statology.org/binomial-probabilities-ti-84-calculator/">BinomCDF</a> command. 
The order of inputs is: n, p, k<pre>n = number of trials = 8
p = probability of success = 0.25
k = number of successes = 3</pre>Type in <font color=red>BinomCDF(8,0.25,3)</font> to get the approximate result <font color=red>0.88618</font>
The answer will vary depending how you round it.


Here are some alternative technology options.<ul><li>Search out "binomial CDF calculator". <a href="https://stattrek.com/online-calculator/binomial">This page</a> and <a href="https://www.gigacalculator.com/calculators/binomial-probability-calculator.php">this page</a> are two of many results. Feel free to explore your favorite.</li><li>Use the <a href="https://geogebra.github.io/docs/manual/en/Probability_Calculator/">Probability Calculator</a> in GeoGebra. Select "binomial" from the dropdown menu. Type n = 8 and p = 0.25; the goal is to calculate *[Tex \large P( 0 \le \text{x} \le 3 )]</li><li>Use the spreadsheet command called <a href="https://support.google.com/docs/answer/3093987?hl=en">BinomDist</a>. The input would be <font color=red>=BinomDist(3,8,0.25,true)</font></li><li>Use the <a href="https://geogebra.github.io/docs/manual/en/commands/BinomialDist/">BinomialDist</a> command in GeoGebra. Note "binomial" instead of "binom". The input would be <font color=red>BinomialDist(8,0.25,3,true)</font></li></ul>Refer to the help manual for more information.


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


<font size=4>The slightly longer method</font>


The binomial probability formula is
B(x) = (nCx)*(p^x)*(1-p)^(n-x)
where,<pre>n = number of trials
p = probability of success
x = number of successes</pre>The nCx refers to the nCr combination formula. These values are found in Pascal's Triangle. A quick way to calculate the nCr values is to use the <a href="https://support.microsoft.com/en-us/office/combin-function-12a3f276-0a21-423a-8de6-06990aaf638a">Combin</a> function in a spreadsheet. Or you can use a <a href="https://education.ti.com/en/customer-support/knowledge-base/ti-83-84-plus-family/product-usage/34718">TI Calculator</a>. 
Like with many things in math, there are many options to calculate nCr.


Let's calculate the probability of exactly 0 police chiefs believe that the death penalty significantly reduces the number of homicides.
B(x) = (nCx)*(p^x)*(1-p)^(n-x)
B(x) = (8Cx)*(0.25^x)*(1-0.25)^(8-x)
B(0) = (8C0)*(0.25^0)*(1-0.25)^(8-0)
B(0) = (1)*(0.25^0)*(1-0.25)^(8-0)
B(0) = 0.10011292
This value is approximate.
It is possible to calculate by hand, given a very long time, but I recommend a calculator. 


Repeat the process to find these other values
B(1) = 0.26696777
B(2) = 0.3114624
B(3) = 0.2076416
I'll skip showing the steps for these. 


Therefore,
B(0)+B(1)+B(2)+B(3)
= 0.10011292+0.26696777+0.3114624+0.2076416
= 0.88618469
= <font color=red>0.88618</font>
The answer will vary depending how you round it.


More practice with the Binomial Distribution is found on <a href="https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1207647.html">this page</a>
</font>