Can complementary events be independent?
The difficulty is caused by the fact that you use the term “complimentary events” when strictly speaking there is no such thing. It should be “complementary OUTCOMES” of the same event. In your example picking a male student or picking a female student are complementary outcomes of the same event, NOT separate events. When you talk about independent events, then you are talking about two separate occurences, each with its own set of outcomes, which is not the same as above. Then, independence means that the outcome of one does not affect the probabilities of the outcomes of the other. This situation arises because strict mathematical language can be overpowering and repetitious so we often use it a little sloppily. We talk about the probability of the event “roll a six with a die” when strictly speaking we should talk about the probability of the “outcome six” from the event “roll a die”. Some people might argue that there is nothing wrong with using the word “event” to mean “outcome”
