SOLUTION: The chance that a U.S. police chief believes the death penalty “significantly reduces the number of homicides” is 1 in 4. If a random sample of 8 police chiefs is selected, fin

Question 1208472: The chance that a U.S. police chief believes the death penalty “significantly reduces the number of homicides” is 1 in 4. If a random sample of 8 police chiefs is selected, find the probability that at most 3 believe that the death penalty significantly reduces the number of homicides.
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

Answer: 0.88618 (approximate)

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

Quick way to find the answer using technology

On a TI83 or similar, you would use the BinomCDF command.
The order of inputs is: n, p, k

n = number of trials = 8
p = probability of success = 0.25
k = number of successes = 3

Type in BinomCDF(8,0.25,3) to get the approximate result 0.88618
The answer will vary depending how you round it.

Here are some alternative technology options.

Search out "binomial CDF calculator". This page and this page are two of many results. Feel free to explore your favorite.
Use the Probability Calculator in GeoGebra. Select "binomial" from the dropdown menu. Type n = 8 and p = 0.25; the goal is to calculate
Use the spreadsheet command called BinomDist. The input would be =BinomDist(3,8,0.25,true)
Use the BinomialDist command in GeoGebra. Note "binomial" instead of "binom". The input would be BinomialDist(8,0.25,3,true)

Refer to the help manual for more information.

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

The slightly longer method

The binomial probability formula is
B(x) = (nCx)*(p^x)*(1-p)^(n-x)
where,

n = number of trials
p = probability of success
x = number of successes

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 Combin function in a spreadsheet. Or you can use a TI Calculator.
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
= 0.88618
The answer will vary depending how you round it.

More practice with the Binomial Distribution is found on this page

RELATED QUESTIONS
The chance that a U.S. police chief believes the death penalty "significantly reduces the (answered by ikleyn)

The chance that a U.S. police chief believes the death penalty “significantly reduces (answered by Boreal)

The chance that a police chief believes that the death penalty will reduce homicides is 1 (answered by oscargut)

The table shows the political affiliation for a random sample of American voters and... (answered by ewatrrr)

Hello, please help me with this question. Thank you in advance In one year, there were (answered by CPhill)

In Canada, 72% of the voters are not in favour of the reinstatement of the death penalty. (answered by Boreal)

It is reported that in the Congress, 60% are in favor of the Death Penalty Law. If a... (answered by Edwin McCravy)

Data from 1988 through 2000 showing the percentage of Americans for and against the death (answered by funmath)

I need help with formulas and how to apply them to this type of question The average... (answered by ewatrrr)