Probability of a conditional in a game with biased coins

Question

The following is a problem in a 2022 logic textbook. I am convinced that the author's solution is incorrect and I would appreciate seeing what answer others might offer. (I will reveal the author's answer and my own answer later.)

Smith's Coins. Smith has 10 coins, each given to a different person, A, B, C,... Five of the coins are double-headed and five are double-tailed. Each person knew the following: If they flip and it lands heads, they win 1.00 dollar; if they flip and it lands tails, they lose 0.50 dollars; nothing happens if they don't flip. Three of the 5 people who received a double-headed coin knew their coin was double-headed, while 2 of the 5 people who received a double-tailed coin knew their coin was double-tailed; the rest thought they were flipping a fair coin. Everyone was rational, risk-neutral, and wanted to win.

Think about person A. Given what we know, we can be certain of the following:

• If A got a double-headed coin and knew it, she flipped and won.
• If A got a double-tailed coin and knew it, she didn't flip and neither won nor lost anything.
• If A thought she had a fair coin, she flipped (since she was rational and flipping maximized her expected utility given her evidence).

Question: What is the probability that "If A flipped, she won"?

More precisely, letting W = "A Won", F = "A Flipped", what is Pr(If F then W) or, equivalently, what is Pr(F-->W)?

Answer 1

Person A flips the coin in 8/10 of cases, which includes the 3/10 of cases where she has a double headed coin and the 5/10 of cases where she doesn't know what coin she has. In all of the 3/10 double headed cases, she gets heads, and in 2/5 of the 5/10 unsure cases does she get heads (since two of the unmarked coins are actually double headed).

There are 8 cases where she flips the coin, and she gets heads in 5 of them. The chance that she wins given that she flips is 5/8. Put another way, all we know from the fact that she flipped is that she's a random person who isn't one of the ones who knows they have a double tailed coin. There are 8 such people, 5 of which have a double headed coin, and 3 of which have a double tailed coin.

Answer 2

This is the author's answer:

"Here, there is a natural intuition that this conditional's probability is equal to the probability that A received a double-headed coin, namely 1/2. After all, she either received a double-headed coin or a double-tailed coin, and if she flipped a double-headed coin, she won, whereas if she flipped a double-tailed coin, she lost."

The author then goes on to point out that "not everyone shares the judgment that [the answer] is 1/2."

But I believe this "intuition" is referring to the unconditional probability that A wins, namely Pr(W) and does not account for the bias towards winning introduced by flipping, since, if A found herself among the 2 people who knew they had losing coins she would not flip. Since 8 people flipped (the 3 who knew they got a winning coin plus the 5 that thought they were flipping utility-maximizing fair coins), and out of those, 5 actually had winning coins, then Pr(F-->W) = 5/8.