The idea/concept of conditional probabilities was not something that I had ever debated. I'm just pointing out that the NBME question doesn't say "which of the following is the probability that he lives the next three years?" It asks, "which of the following is the probability that he will survive 4 years?" And as I said before, if you flip one tails, the odds of landing three more are 1/8, but if you flip one tails, the odds of landing four are still 1/16. I understand what NBME is trying to ask, but the grammar doesn't support that.
You conveniently left out an important part of the question, it actually says: "based on this information, if a patient survives for 1 year, which of the following is the probability that he will survive for 4 years"
Also, as everyone else has been saying, if I flip one coin and get tails, the probability of getting tails again the next time is still 50%. The act of my previous coin flipping does not make more or less probable of getting heads or tails, the events are absolutely independent.
Now, if you were to say, what is the probability of flipping a coin 3 times in a row and getting tails each time, then it would be as you say, 1/8. However, if I then say, given that I have already flipped the coin 3 times, what is the probability that my 4th flip will be tails? This time I have specified my previous coin flippage, and thus any future coin flips are independent of what I just did; therefore, it will be a 50% chance of getting tails, given the information that I already flipped the coin 3 times and got tails.
In contrast, if I ask the question, what is the probability of flipping a coin 4 times in a row and getting tails each time, then the probability is, again as you say, 1/16. Again, this is much different if I say that I flipped a coin 3 times and got tails each time, what is the probability of getting tails on the next flip (i.e., the fourth flip), the answer is 50%.
The only time you can multiply probabilities the way you are suggesting is when you are predicting an outcome, not if something has already occurred. As people have pointed out, you are speaking exactly as gambling addicts speak; i.e., I bet my money on black and lost the last 10 times; therefore, I am more likely to win on black on the next round. This fallacy is due to forgetting that what happened in the past cannot influence any future events.
In contrast, if I am a gambler and I sit down at a table and I am trying to plan how I am going to lose my money, the accurate way of thinking would be as follows, "hmmm.. whats the probability that I will lose if I bet black 10 times in a row? ah, 1/20; therefore, it is highly unlikely to lose on black if I bet 10 times in a row!!!"
Now, the 50 year-old patient asks, "what is the probability I will survive to the age 54 doc?", well, then the answer is 0.8*0.875*0.9*0.9 = 0.567. This is because the patient has not begun to encroach on the first probability
If the patient comes back next year, and is still alive, and the he says, "You told me last time my probability to live to be 54 years was 0.567, I already lived one year, what is my probability of still living to 54 years old?" Then, you tell the patient, oh since you already lived that one year, your probability of living to 54 (i.e., 3 more years) is 0.875*0.9*0.9 = 0.7088
The KEY is that once an event has already occurred, it cannot influence the future event. The likelihood of flipping a tails 4 times in a row is 1/16, but the likelihood of flipping a tails 4 times in a row, knowing that I already received a tails on the first flip, is 1/8 since any future event is independent of the knowledge of the previous flip.
I feel like I'm rambling now