If you assume a unique classical universe with one-way causality, then obviously, the fact you only throw heads is just luck, it doesn’t say anything about the initial throw. You could promote hypothesis like “the coin is biased, because the ones running that experiment are sadistic who like to see my fear every time the coin is tossed”, and maybe from that you could change your estimate of the first coin being biased, depending of your understanding of human psychology.
If you assume MWI it gets more complicated, and depends if the coin tossing is “quantum-random” or not. A usual coin tossing is not “quantum-random”, the reasons behind the coin ending heads or tails are in classical physics, well above the level of “quantum noise”. So in all the existing worlds (or a very, very large majority of them), the toss will give the same results, so you’re back to the first case.
If you assume MWI and the coin toss are “quantum random”, then after two tosses (not counting the initial), you’ve 8 outcomes : HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. You can only experience H and THH. On the 8 copies of you, 3 will be dead, 5 alive. But for there are two copies seeing HH, you can’t tell apart THH and HHH, both have p=1/2. p(HH | I’m alive) = 2⁄5, but p(T | *HH and I’m alive) is still 1⁄2.
Now, if you assume the first coin was not quantum, but the subsequent are … there is no longer 8 worlds having the same Born probabilities (the same “level of existence”) but only two sets of 4 worlds, each set having a 1⁄2 probability of existing. In one set of world, there will be just one copy of you seeing “HH”, on the other, there will be 4 copies of you seeing all the possible outcomes.
Then we can rephrase the problem in a way that doesn’t involve the anthropic principle, making it a more classical problems of probabilities. We take 5 similar pieces of paper. Two are written HH, the others HT, TH and TT. One HH is put aside, then each are folded. Then someone tosses a coin. If it’s tails, you are given the lone HH paper. If it’s heads, you’re giving one at random among the 4 papers. You open your paper, and it’s HH. What’s the probability the coin landed tails ? Well, it’s 4⁄5, not 1⁄2.
So, my answer is : if MWI holds, and the first coin is not quantum random, but the later are quantum random, you should consider the 1000 heads (and even stop much before) to be strong evidence towards “the initial was tails”. If none or all of the coins are quantum random, or you don’t believe in MWI, you shouldn’t.
Let’s took at the different possible assumptions.
If you assume a unique classical universe with one-way causality, then obviously, the fact you only throw heads is just luck, it doesn’t say anything about the initial throw. You could promote hypothesis like “the coin is biased, because the ones running that experiment are sadistic who like to see my fear every time the coin is tossed”, and maybe from that you could change your estimate of the first coin being biased, depending of your understanding of human psychology.
If you assume MWI it gets more complicated, and depends if the coin tossing is “quantum-random” or not. A usual coin tossing is not “quantum-random”, the reasons behind the coin ending heads or tails are in classical physics, well above the level of “quantum noise”. So in all the existing worlds (or a very, very large majority of them), the toss will give the same results, so you’re back to the first case.
If you assume MWI and the coin toss are “quantum random”, then after two tosses (not counting the initial), you’ve 8 outcomes : HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. You can only experience H and THH. On the 8 copies of you, 3 will be dead, 5 alive. But for there are two copies seeing HH, you can’t tell apart THH and HHH, both have p=1/2. p(HH | I’m alive) = 2⁄5, but p(T | *HH and I’m alive) is still 1⁄2.
Now, if you assume the first coin was not quantum, but the subsequent are … there is no longer 8 worlds having the same Born probabilities (the same “level of existence”) but only two sets of 4 worlds, each set having a 1⁄2 probability of existing. In one set of world, there will be just one copy of you seeing “HH”, on the other, there will be 4 copies of you seeing all the possible outcomes.
Then we can rephrase the problem in a way that doesn’t involve the anthropic principle, making it a more classical problems of probabilities. We take 5 similar pieces of paper. Two are written HH, the others HT, TH and TT. One HH is put aside, then each are folded. Then someone tosses a coin. If it’s tails, you are given the lone HH paper. If it’s heads, you’re giving one at random among the 4 papers. You open your paper, and it’s HH. What’s the probability the coin landed tails ? Well, it’s 4⁄5, not 1⁄2.
So, my answer is : if MWI holds, and the first coin is not quantum random, but the later are quantum random, you should consider the 1000 heads (and even stop much before) to be strong evidence towards “the initial was tails”. If none or all of the coins are quantum random, or you don’t believe in MWI, you shouldn’t.