This is my favorite of the batch you asked about.
There is definitely a flaw in this argument. John is looking to win a particular raffle (the Mayfield one). Let's say his odds are 1 in 100 (1% chance).
Now, let's say there are four other raffles, each with a 1 in 100 chance (1%) of winning. If John enters all five raffles, he has a better chance of winning SOMETHING. (Mathematically, roughly 5% chance of winning at least one of those raffles.) However, if he does win something, it doesn't have to be the Mayfield prize he wants. He could win something else.
The other raffles are a red herring. He may have a better chance of winning SOMETHING, but the odds of winning the Mayfield prize specifically is still 1 in 100. That doesn't change. If he wants a better chance of winning the Mayfield prize, he'd have to enter THAT raffle more than once.
The lottery question you refer to is a similar flaw. Let's say your chances of winning a lottery are 1 in 1,000,000. Every person who plays has the same chance. If you play by yourself -- your odds of winning are 1 in 1,000,000. If you play with thousands of other people, your odds are still 1 in 1,000,000. You're not playing against THEM -- you're still trying to guess what random numbers are going to come up. Other people playing don't affect that.
Now, if you entered a raffle all by yourself as opposed to entering a raffle with thousands of other people, that's a different story!!