Question 1201405
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The rules of this website is to post one question at a time. 
I'll do problem 1 to get you started.


We have a binomial process because:
<ul><li>A person gets a headache or they don't. There are two outcomes.</li><li>Each person is independent of any other (eg: someone getting a headache won't cause another person to get a headache)</li><li>The probability of a headache is the same for any person (10%).</li></ul>n = 5 = sample size
p = 0.10 = probability of a headache
x = number of people, in the sample, that get a headache
x is an integer in the set {0, 1, 2, 3, 4, 5}
i.e. x is between 0 and 5 inclusive of both endpoints.


B(x) = binomial probability
B(x) = (n C x)*(p^x)*(1-p)^(n-x)
B(x) = (5 C x)*(0.10^x)*(1-0.10)^(5-x)
B(x) = (5 C x)*(0.10^x)*(0.90)^(5-x)
The nCx refers to the nCr combination formula.


Plug in x = 1 to determine the probability there's exactly one person with a headache in the sample of five people.
B(x) = (5 C x)*(0.10^x)*(0.90)^(5-x)
B(1) = (5 C 1)*(0.10^1)*(0.90)^(5-1)
B(1) = (5)*(0.10^1)*(0.90)^4
B(1) = <font color=red>0.32805</font>


The probability that there's exactly one headache is exactly <font color=red>0.32805</font>
This is exactly a 32.805% chance.


Here are a few binomial distribution calculators.
<a href = "https://www.gigacalculator.com/calculators/binomial-probability-calculator.php">https://www.gigacalculator.com/calculators/binomial-probability-calculator.php</a>
<a href = "https://www.omnicalculator.com/statistics/binomial-distribution">https://www.omnicalculator.com/statistics/binomial-distribution</a>
You could also use a spreadsheet or a TI83/TI84 calculator.


Here is an article talking about binomial probabilities using a TI84
<a href = "https://www.statology.org/binomial-probabilities-ti-84-calculator/">https://www.statology.org/binomial-probabilities-ti-84-calculator/</a>
Use the PDF rather than CDF because the goal is to calculate one single value (rather than add up a bunch of values).


Another question involving the binomial probability distribution
<a href = "https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1201351.html">https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1201351.html</a>


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Answer: <font color=red>0.32805</font>
This value is exact and hasn't been rounded.
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