Chiropteran, just on the off chance that you're not trolling, I offer for you two different (but closely related) scenarios.
One: You have two coins. One is H/T, the other is H/H. (Let's label the coins A and B, thus you have heads on A (HA), tails on A (TA), the first head on B (H1B) and the 2nd head on B (H2B). I mix them up thoroughly and randomly select one side of a coin to show you. HA, TA, H1B, and H2B are all equally likely, so there's a 1/4 chance that you're seeing any of those sides. Here's the puzzler: If you see heads, what's the probability that the other side is also heads? Well, if you saw a tails, you'd know exactly which side you were looking at, and you'd answer that there's a 100% probability that the other side is heads. But, you're not looking at a tails, you're looking at a heads. You know that there are 3 heads, and each is equally as likely as the others to be the side that you're looking at. Intuitively, most people would say "50%" because you could have either coin, and each coin has equal probability. Thus, what you're looking at might be HA, it might be H1B, and it might be H2B. Each of those 3 has an equal probability of occurring. And, for 2 out of 3 of them, the other side is a head. Only one out of 3 (HA), the other side is a tails. So, if you have those two coins and see only one side of one coin, and it's heads, what's the probability that the other side is heads? The answer is 67%
Scenario Two: You have 20 100-sided dice. 19 of those dice have the sides numbered from 1 to 100. The 20th die has "50" on every side. You randomly reach into a bag and select one of those 20 dice. You roll it and are only able to observe the side that comes up. You see a "50". You think to yourself, "well, it could have been any of the dice."
You roll it again and get "50" a 2nd time. Again, you think to yourself, "well, that doesn't prove anything. It could be any of the dice."
You roll it again and again and again, 10 times in a row. Each time you get a "50". You're a little slow in the head (apparently), so you roll it again and again and again. Finally, around about the 60th roll (would it really take you this long?), you finally remark "you know, I think that I probably have the die that has a 50 on every side. You keep rolling again and again and again. On the 120th roll, someone asks you "what do you think the probability is that your next roll is a 50? Hopefully, your answer would sound like this "I'm pretty damn sure that I've got the die that has a 50 on every side. I can't absolutely guarantee a 50 on the next roll, because there's an incredibly unlikely chance that I just got lucky and rolled a 50 119 consecutive times on a 100 sided die numbered from 1 to 100, but I'd guess the probability of getting a 50 on the next roll is damn close to 100%."
Now, see that "probably" that I bolded (underlined and italicized) for you? A lot of us can calculate the exact probability of that probably. Starting with the 1st role, and every roll after that, we can calculate the probability that you were holding the "50" die. (Starting before you rolled, the probability of having the "50" die was 1 in 20. The first time you rolled and got a "50", a smart person would be thinking "you know, there's a pretty good chance that I've got the die with 50 on every side." If it helps, imagine that instead of 100 sided dice, they are 11 billion sided dice, one of which has 50 on every side. It should be pretty intuitive in that case that on the first roll, if you got a 50, you can be almost certain that you have the die with 50 on every side.
Now, back to the original problem:
One of the coins is a trick coin that has both sides heads. You pick a random coin from the jar and flip it 4 times, and each time it comes up heads. What are the chances that the next flip will come up heads?
Hopefully that no longer seems ambiguous. It's not.
Edit: and remember, in each of those alternate problems I posted, you picked the die/coin at random.
Your argument:
"There are 3 jellybeans in a jar: a red one, a yellow one, and a green one. I select one of the jellybeans, completely at random, then look at it. OMFG, it's red!"
You: "nuh uh, it's not randomly selected if it was red."