65% of the students in a certain small college's population are women. It is also noted at this college that women are twice as likely to take poetry as men. In one semester 20% of the college's women take poetry. If a randomly selected college student is in a poetry class, what is the probability that this student is a woman?
I know this looks a bit complicated at first glance, and there are two ways to interpret the problem, but let's choose the most likely interpretation and sort through some of the numbers. I'm assuming that this problem is NOT implying that one-third of the poetry students are men. It does say, "women are twice as likely to take poetry," but some of the information given does not support the idea that there are two women for every man in poetry classes. It seems more likely to mean that "any particular woman" is twice as likely to take poetry than "any particular man." In other words a smaller population of men are INDIVIDUALLY half as likely to choose poetry classes, while a larger population of women are INDIVIDUALLY more likely to choose poetry classes. I predict that there will be several more than 2 women for every man who chooses a poetry class.
If "20% of the college's women take poetry," and 65% of the college's students are women, then 13% of the college's population are women who take poetry (65% X 20% = 65%/5 = 13%). Since women are "twice as likely" as men to take poetry, 10% of the college's men take poetry (20%/2 = 10%). We know that 35% of the college's population is men (100% - 65% = 35%), so 3.5% of the college's population are men who take poetry (35% X 10% = 35%/10 = 3.5%). Unfortunately, we do not know how many students attend the university or take poetry classes, but we should be able to find proportions of men and women who make up poetry classes and use those proportions to determine the likelihood that a woman is randomly selected.
Let's divide the percentage of women who take poetry by the percentage of men who take poetry. Thus, 13%/3.5% = 3.7143. This means that for every man who takes poetry at this college, 3.7143 women take poetry. So, it follows that for every 4.7143 students in a poetry class, 1 of them is a man. We can convert this to a percentage: 21.212% of poetry students are men (100/4.7143), while 78.788% are women (100% - 100/4.7143). Now, here's the fun part: if a student at the college is selected randomly from the entire student population - AND they are found to be taking a poetry class - there is a 78.788% chance that they are a woman. I hope this helps!