Question 1199907
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I'll focus on part (c) only.


For questions like this, you could calculate by hand. 
However that is tedious busy-work in my opinion. The reason why I say this is because you would have 16 separate calculations, each of which are somewhat long in their own right.


It's better to use a specialized binomial distribution calculator such as this one
<a href = "https://www.gigacalculator.com/calculators/binomial-probability-calculator.php">https://www.gigacalculator.com/calculators/binomial-probability-calculator.php</a>
In this case:
n = 45 = sample size
p = 0.66 = probability of a repeat offender
x = 30 = number of repeat offenders
Look for where it says "Probability of X ≥ 30 events".



If you want to use your TI83 or TI84 calculator, then refer to this page for more information
<a href = "https://www.statology.org/binomial-probabilities-ti-84-calculator/">https://www.statology.org/binomial-probabilities-ti-84-calculator/</a>
Focus on the binomialcdf function and NOT the binomialpdf.
That page explains how to find the binomialcdf function. Also there are examples.


Something like 
binomialcdf(45,0.66,29)
will calculate the sum of the probability values from x = 0 to x = 29


Subtract that from 1 to find the sum of the probabilities from x = 30 to x = 45.


This is valid because
P(0 ≤ X ≤ 29) + P(30 ≤ X ≤ 45) = 1 


Therefore you'll type into the TI83 or TI84 calculator <font color=red>1-binomialcdf(45,0.66,29)</font> to find the probability of having at least 30 repeat offenders.
Be careful to avoid typing in 1-binomialcdf(45,0.66,30) as that would slightly be off.


If instead you want to use a spreadsheet, then the command is called BINOMDIST
<a href = "https://support.microsoft.com/en-us/office/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c">https://support.microsoft.com/en-us/office/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c</a>


There are many other options.


Let me know if you have any questions about a particular piece of software, website, spreadsheet, or calculator. 


Hint about the final answer: It is between 0.5 and 0.6
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