## For question 3, let's think about the expected gain.

If you buy two tickets from one lottery, the probability of winning any specific number is the same for both tickets. The expected gain from those two tickets can be calculated by multiplying the probability of winning by the payoff for each ticket and summing them up.

On the other hand, if you buy one ticket from each lottery, the probability of winning any specific number is also the same for both lotteries. The expected gain from those two tickets can also be calculated by multiplying the probability of winning by the payoff for each ticket and summing them up.

To determine which option is better in terms of expected gain, you will need to compare the two calculated values.

Now, for question 4, we need to find the expected area of a random square.

To solve this, we'll first note that the area of a square is given by the formula A = side length * side length.

The side length of the square is a uniform random variable with a range of [0,1]. Since it is a uniform random variable, the probability density function (pdf) is constant over this range.

To find the expected area, we need to calculate the integral of the area formula (A = side length * side length) multiplied by the pdf of the side length.

The integral would be from 0 to 1, and the result will give us the expected area of the square.

Finally, for question 5, we need to find the expected number of matches when n men throw their hats into a pile and each man takes a hat at random.

Let's define a success as a man picking his own hat. If a man picks his own hat, we consider it a match. If he doesn't pick his own hat, we consider it a failure.

The probability of success for each man is 1/n, as there are n hats in total and each hat is equally likely to be chosen by any man.

We can express the number of matches as a sum of n Bernoulli random variables, where each variable represents whether a specific man picked his own hat.

The expected number of matches can be calculated by taking the sum of the expected values of these n Bernoulli random variables.

The expected value of each Bernoulli random variable is given by the probability of success, which is 1/n.

By taking the sum of these n probabilities, we can find the expected number of matches.

I hope this helps you understand the methods behind solving these questions! Let me know if you have any further questions.