Okay, let's try to figure out the probability that both Dan and Freddie win their respective races. So, there are four candidates in total: Dan, Donald, Freddie, and Bernie. Each of them has a 50% chance of winning their own elections for president and vice president. The question is asking for the probability that both Dan (president) and Freddie (vice president) win. 

First, I need to understand the problem clearly. We have two separate elections: one for president and one for vice president. Each election has two candidates, and each has an equal chance of winning. So, for the president election, Dan and Donald are running, each with a 50% chance. Similarly, for the vice president election, Freddie and Bernie are running, each with a 50% chance. The events of Dan winning the president election and Freddie winning the vice president election are independent because the outcome of one doesn't affect the outcome of the other. 

Wait, but the problem states that each candidate has a 50% chance of winning. Does that mean each candidate in their own election has a 50% chance? So, in the president election, Dan has a 50% chance, Donald 50%. In the vice president election, Freddie 50%, Bernie 50%. Since these are separate elections, the winners are independent. Therefore, the probability that both Dan and Freddie win is the product of their individual probabilities. 

So, the probability that Dan wins president is 1/2, and Freddie wins vice president is also 1/2. Since these are independent events, we multiply the probabilities: 1/2 * 1/2 = 1/4. 

But wait, let me double-check. Are there any dependencies or other factors I'm missing? For example, is there any chance that the candidates could influence each other's chances? The problem states that each candidate has a 50% chance of winning. It doesn't mention any dependencies or external factors affecting the probabilities. Therefore, it's safe to assume that the events are independent. 

Another way to think about it: list all possible outcomes. There are two candidates for president and two for vice president, so the total number of possible outcomes is 2 * 2 = 4. Each outcome is equally likely. The possible outcomes are:
1. Dan president, Freddie VP
2. Dan president, Bernie VP
3. Donald president, Freddie VP
4. Donald president, Bernie VP

Out of these four equally likely outcomes, only one outcome has both Dan and Freddie winning. Therefore, the probability is 1/4. 

Alternatively, using probability rules: the probability of both independent events happening is the product of their probabilities. Since each has a 1/2 chance, multiplying them gives 1/4. 

I think that's solid. Let me see if there's another angle. Suppose someone might confuse the total number of candidates, but the problem specifies each of the two elections (president and vice president) has two candidates each. So, the independence is between the two elections, not within them. 

Another possible mistake could be considering that all candidates have 50% chance, but since each election is separate, it's just 1/2 for each. So, combining them as independent gives 1/4. 

Yes, I believe that's correct. Therefore, the probability that both Dan and Freddie win is 1/4.
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