Question 1201405: If 10% of the people who take a certain medication get a headache, find the probability that if 5 people take the medication, one will get a headache.
If 25% of U.S. prison inmates are not U.S. citizens, what is the probability of randomly selecting three inmates who will not be U.S. citizens?
If 30% of commuters ride to work on a bus, find the probability that if 8 workers are selected at random, 3 will ride the bus.
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
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:
- A person gets a headache or they don't. There are two outcomes.
- Each person is independent of any other (eg: someone getting a headache won't cause another person to get a headache)
- The probability of a headache is the same for any person (10%).
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) = 0.32805
The probability that there's exactly one headache is exactly 0.32805
This is exactly a 32.805% chance.
Here are a few binomial distribution calculators.
https://www.gigacalculator.com/calculators/binomial-probability-calculator.php
https://www.omnicalculator.com/statistics/binomial-distribution
You could also use a spreadsheet or a TI83/TI84 calculator.
Here is an article talking about binomial probabilities using a TI84
https://www.statology.org/binomial-probabilities-ti-84-calculator/
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
https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1201351.html
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Answer: 0.32805
This value is exact and hasn't been rounded.
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