If $P(E)=0.55, P(E$ or $F)=0.80$, and $P(E$ and $F)=0.05$, find $P(F)$.

Step 1 ：We are given that the probability of event E, denoted as \(P(E)\), is 0.55.

Step 2 ：We are also given that the probability of either event E or event F occurring, denoted as \(P(E \text{ or } F)\), is 0.8.

Step 3 ：Additionally, we are given that the probability of both event E and event F occurring, denoted as \(P(E \text{ and } F)\), is 0.05.

Step 4 ：We know that the probability of either event E or event F occurring is equal to the sum of the probabilities of event E and event F occurring individually, minus the probability of both event E and event F occurring. This is because the event of both E and F is counted twice when we add the probabilities of E and F, so we need to subtract it once to correct this.

Step 5 ：Using this information, we can solve for the probability of event F, denoted as \(P(F)\).

Step 6 ：Substituting the given values into the equation \(P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F)\), we get \(0.8 = 0.55 + P(F) - 0.05\).

Step 7 ：Solving this equation for \(P(F)\), we find that \(P(F) = 0.3\).

Step 8 ：Final Answer: The probability of event F occurring is \(\boxed{0.3}\).