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A man tells me he can perfectly predict a coin toss. I think the probability of that being true is $P(A) = 2^{-16}$, where A is the event that the man is psychic and can perfectly predict the coin toss. If not, the coin is fair and the man has a 50/50 chance. $C_k$ denotes the event where coin toss number $k$ is predicted correctly.

What do I think the probability that the man can predict the first two coin tosses is?

So for the first coin toss I thought if the man is psychic, he will 100% predict the coin, if not, he has a 50% chance, so: $$ P(C_1) = P(A)\cdot 1+(1-P(A))\cdot0.5 $$ I would think that $C_1$ and $C_2$ are independent and the probability of predicting a coin is the same, no matter what toss it is, so $P(C_1 \cap C_2) = P(C_1)\cdot P(C_2) = P(C_1)^2$.

But I can also think like this, if the man is psychic, he has 100% chance of guessing correct twice, otherwise the chance is $0.5^2$: $$ P(C_1 \cap C_2) = P(A) + (1-P(A))\cdot0 .5^2 $$

These two are not equal so I am wondering where I go wrong

Matty111
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    $P(C_2)$ will equal $P(C_1)$ which you should note is not exactly equal to $0.5$. $P(C_2\mid C_1^c)=0.5\neq P(C_2)$ since knowing he was wrong about the first coin proves he is not psychic and that he is guessing randomly. This proves that $C_2$ and $C_1$ are in fact dependent events, not independent. – JMoravitz Jan 22 '21 at 15:50
  • Thanks @JMoravitz. So $C_1$ and $C_2$ are both depedent on A, because they share this, they are dependent on each other? If the man tosses 2 coins at the same time, would the answer be different? – Matty111 Jan 22 '21 at 16:34
  • "because they share this..." No, that is irrelevant. You can have events $E,F,G$ with $E$ and $F$ both dependent on $G$ while $E$ and $F$ are independent of each other. "If the man tosses 2 coins at the same time..." No, the answer will be the same. The only logical way to model the problem is to have the coins be distinct, whether distinguishable to our eyes or otherwise. The problem knows the coins are different even if we personally do not. – JMoravitz Jan 22 '21 at 16:37
  • So, the problem can say for instance "The coin that was minted in 1987 landed heads while the coin that was minted in 1993 landed tails" even though we might not have ever thought to look at what year the coins were minted. The coins physically occupy different positions in space. In treating the coins as distinct, this allows us to better organize our sample space for correct calculations. See more here. – JMoravitz Jan 22 '21 at 16:40

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