I have a solved question from Ross as stated below.
Q : Suppose that each of three men at a party throws his hat into the center of the room. The hats are first mixed up and then each man randomly selects a hat. What is the probability that none of the three men selects his own hat?
Sol: We shall solve this by first calculating the complementary probability that at least one man selects his own hat......
I want to start with basics and identify the sample space first.
$$\mathrm{Space} = \{h_1p_1, h_1p_2, h_1p_3, h_2p_1, h_2p_2, h_2p_3, h_3p_1, h_3p_2, h_3p_3\}$$
where $h_ip_j$ is the event of picking up a hat of person $i$ by person $j$.
To satisfy the condition that nobody picks his own hat, $i$ should not be equal to $j$.
Why the complement is "at least one man selects his own hat" and not that "all guys select their own hat".