Let's say, at a certain university, over the last few years 100 students have graduated with a degree in Computer Science, and another 100 in Biology. Let's also say 60 of the CS graduates found a job within 3 months of their graduation, and 50 of the Bio graduates did.
The question is: what is the probability that a student from either (as in "at least one of)) programme gets a job within 3 months of graduation?
The Attempt at a Solution
First, I tried adding the groups together, saying the probability is 0.5 for Bio and 0.6 for CS, and slapped them together thinking the two are independent events, but this gave me a probabality of 1.1, which can't be right. So then I thought maybe the probability should be (50 + 60) / (100 + 100), which is 0.55, and this made sense to me because as a matter of fact, 110 out of 200 students did find a job within 3 months of graduation.
But then someone suggested the answer should be 1 minus the probability that none of them get a job, which would be 1 - ((1- 0.60) * (1- 0.50)) = 1 - (0.40 * 0.50) = 0.80 != 0.55. And then I thought maybe my first approach had overlooked the possibility of graduates from both programmes finding a job. But adding the following together:
The probability of a CS grad but not the Bio one finding a job 0.60 * 0.50
The probability of a Bio grad but not the CS one finding a job 0.40 * 0.50
gives 0.30 + 0.20 = 0.50 != 0.55.
I'm really confused here. What am I doing wrong? I seem to have misunderstood something very fundamental.