# Conditional probability of positive response in a survey

A college’s student population consists of 29% freshmen, 27% sophomores 25% juniors and 19% seniors.

The school’s transportation department wants to expand their service but to do this a fee will be charged to the students. A survey done by the school determined that 68% of the freshmen, 61% of the sophomores, 42% of the juniors and 35% of the seniors agree with the extra fee to improve the service.

(a) What is the probability that a randomly chosen student agrees to pay the fee?

(b) If a student agrees to pay the fee, what is that probability that he or she is a senior?

for (a) I multiply each population percentage by the percentage of students in that population that agree and took the union. The result comes up to be 0.553

for (b) I'm trying to use P(senior|agrees to pay) but I keep getting the result 0.0665 which is just senior population multiplied by the seniors percentage that agreed, which seems wrong. Any ideas on (b) would be greatly appreciated

• You know $P(\text{agrees to pay} \mid\text{senior})$. Now use Bayes' theorem. – Arthur Jun 21 '15 at 14:28
• is that simple? – Joz Jun 21 '15 at 15:26

Beginners often find the full form of Bayes' Theorem confusing, specially when only symbols rather than words are used, but the simple form works as well.

P(senior | agrees to pay) = P(senior ∩ agrees to pay)/P(random student agrees to pay)

You have already computed both the values for the right hand side. Proceed ...

• Got it, I did use Baye's rule and it makes sense. Thanks – Joz Jun 21 '15 at 15:59