Independent & dependent probability

The marching band is holding a raffle at a football game with two prizes. After the first ticket is pulled out and the winner determined, the ticket is taped to the prize. The next ticket is pulled out to determine the winner of the second prize. Are the two events independent? Explain. Now before we even think about this exact case, let's think about what it means for events to be independent. It means that the outcome of one event doesn't affect the outcome of the other event. Now in this situation, the first event-- after the first ticket is pulled out and the winner determined-- the ticket is taped to the prize. Then the next ticket is pulled out to determine the winner of the second prize. Now, the winner of the second prize-- the possible winners, the possible outcomes for the second prize, is dependent on who was pulled out for the first prize. You can imagine if there's three tickets, let's say there's tickets A, B, and C in the bag. And for the first prize, they pull out ticket A. That's for the first prize. Now, when we think about who could be pulled out for the second prize, it's only going to be tickets B or C. Now the first prize could have gone the other way. It could have been A, B, and C. The first prize could have gone to ticket B. And then the possible outcomes for the second prize would be A or C. So the possible outcomes for the second event, for the second prize, are completely dependent on what happened or what ticket was pulled out for the first prize. So these are not independent events. The second event-- the outcomes for it, are dependent on what happened in the first event. So they are not independent. The way that we could have made them independent is, after the first ticket was pulled out, if they just wrote down the name or something, and then put that ticket back in. Instead they taped it to the prize. But if they put that ticket back in, then the second prize, it would have still had all the tickets there. It wouldn't have mattered who was picked out in the first time because their name was just written down, but their ticket was put back in. And then you would have been independent. So if you had replaced the ticket, you would have been independent. But since they didn't replace the ticket, they taped it to the prize, these are not independent events.