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)
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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, kn = 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.
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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
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