# In a fictional stats class, 40% of students are female, and the rest are male. Of the female students, 30% are less than 20 years old and 90% are less than 30 years old. Of the male students, half are less than 20 years old and 70% are less than 30 years old. Make a contingency table to describe these two variables Find the probability that a randomly selected studet is 30 years or older If a student is 20 years or older, what is the probability that the student is female? If a student is less than 30 years old, what is the probability that the student is 20 years or older?

Diego Barr 2022-10-20 Answered
Problem:
In a fictional stats class, 40% of students are female, and the rest are male. Of the female students, 30% are less than 20 years old and 90% are less than 30 years old. Of the male students, half are less than 20 years old and 70% are less than 30 years old.
(a) Make a contingency table to describe these two variables
(b) Find the probability that a randomly selected studet is 30 years or older
(c) If a student is 20 years or older, what is the probability that the student is female?
(d) If a student is less than 30 years old, what is the probability that the student is 20 years or older?
My Thoughts:
(b) P(<30 years) = 1 - 0.78 = 0.22
(c) What I first did was find P(S2 given 'not A1'), but the answer doesn't make sense because the denominator ended up being smaller than the nominator.
(d) Do I solve this problem by doing 'not 20 years'?
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bigfreakystargl
Let F denote female; let M denote male; let A denote age.
Since 40% of the students are female and 30% of them are less than 20 years old, the probability that a student is female and less than 20 years old is

Since 90% of the female students are less than 30 years old, the probability that a student is female and less than 30 years old is

The probability that a student is female, at least 20 years old, and less than 30 years old can be found by subtracting the probability that she is less than 20 years old from the probability that she is less than 30 years old, which yields

Finally, the probability that a student is female and at least 30 years old is found by subtracting the probability that a student is female and less than 30 years old from the probability that a student is female, which yields

By using similar reasoning, we can fill in the table for the male students.
$\begin{array}{lcccc}& A<20& 20\le A<30& A\ge 30& Total\\ F& 0.12& 0.24& 0.04& 0.40\\ M& 0.30& 0.12& 0.18& 0.60\\ Total& 0.42& 0.36& 0.22& 1\end{array}$
The probability that a student is at least 30 years old is stated in the contingency table.
To find the probability that a student who is at least 20 years old is female, divide the probability that a female student is at least 20 years old by the probability that a student is at least 20 years old, both of which can be found by adding the appropriate columns in the table.
The probability that a student who is less than 30 years old is at least 20 years old can be found by subtracting the probability that the student is less than 20 years old from the probability the student is less than 30 years old. To find the probability that a student is less than 30 years old, you can subtract the probability that a student is greater than 30 years old from 1.
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