Secret Wins

A group of 100 students are participating in a coin-tossing challenge at a school event. Each student flips a fair coin once: - If the coin lands on heads, the student wins and truthfully reports their win. - If the coin lands on tails, the student flips again: - If the second flip is heads, the student lies and claims they won, even though they lost. - If the second flip is tails, the student truthfully reports their loss.

At the end of the game, 30 students report that they won. As an analyst, you are tasked with determining the expected number of students who actually won the game (i.e., got heads on their first flip).

Example:

Suppose only 8 students report winning. How many students would you expect to have truly won based on the rules above?

Note: You may assume each coin flip is independent and fair (probability of heads = 0.5).