I just took a probability final exam and was fairly confident in my solution of 28/31, but I wanted to be sure... because according to http://www.stat.tamu.edu/~derya/stat211/SummerII02/Final.Summer02.doc which has it as the second question, the answer is .6627. What's discerning is that they have the decimal equivalent of 28/31 as one of their answers which makes it seem like they know something I don't...
"Seventy percent of all cattle are treated by an injected vaccine to combat a serious disease. The probability of recovery from the disease is 1 in 20 if untreated and 1 in 5 if treated. Given that an infected cow has recovered, what is the probability that the cow received the preventive vaccine?"
Here's my solution: Let A be the event a cow recovered, let B be the event a cow received the vaccine.
We are given:
P(A|B) = 1/5 P(A|~B) = 1/20 P(B) = 7/10
We want to find P(B|A), so use Bayes' rule and the total probability theorem to find
P(B|A) = P(A|B) x P(B) / (P(A|B) x P(B) + P(A|~B) x P(~B) ).
Plugging in the values from what's given above, we get (.2 x .7) / (.2 x .7 + .05 x .3) which gives 28/31.
If I'm wrong, I'd love to be pointed in the right direction haha