Okay, lets write down what we do know...
4 out of the 12 cups are decaf, so the odds of selecting a decaf cup by random is 4/12 or 1/3
Alternatively 8 of the 12 cups are caffeinated, so the odds of selecting a caffeinated cup by random is 8/12 or 2/3
P(decaf)=1/3
P(caf)=2/3
But those odds are only for the first cup picked. The second cup has 4 probabilities...
1. Picking a decaf cup for the second time
2. Picking a decaf cup for the first time
3. Picking a caf cup for the second time
4. Picking a caf cup for the first time
So lets find the probability of each event...
1. Picking a decaf cup for the second time. If a decaf cup has already been picked, then the odds of drawing another one is 3/11
2. Picking a decaf cup for the first time. If a decaf cup hasn't already been picked, then the odds of drawing one is 4/11
3. Picking a caf cup for the second time. If a caf cup has already been picked, then the odds of drawing one is 7/11
4. Picking a caf cup for the first time. If a car cup hasn't already been picked, then the odds of drawing one is 8/11.
Okay, so now we can solve for the various possibilities.
1. Picking a decaf cup for the second time
1/3*3/11=3/33
2. Picking a decaf cup for the first time
2/3*4/11=8/33
3. Picking a caf cup for the second time
2/3*7/11=14/33
4. Picking a caf cup for the first time
2/3*8/11=16/33
Both cups were caffeinated, so neither were decaf.
Answer= Neither cup is decaf.
