Two cards are drawn from the deck without regard to order, so the sample space  is the set of all combinations of two cards from a set of nine. The size of this sample space is 

The event is the set of all combinations of two cards from the set of four red cards. The size of this event space is

The probability of the event is

This problem can be modeled using a Markov chain matrix. If we let the rows represent the initial party affiliation, the columns represent final party affiliation, and Democrats be represented by the first row and column, the Markov matrix that represents the system is

The initial state of the Markov system can be represented by the row vector

The distribution of Democrats and Republicans after five years can be determined by evaluating the product

.

Using a calculator, we can determine that this is

The second number will be larger, so there will be more Republicans. Also, the difference will be

.

Republicans will outnumber Democrats by about 300.

If a payoff matrix has a saddle point, it must be both the minimum element of its row and the maximum element of its column. The minimum element of each row is shown in red:

Of these three entries, only 75 is the maximum of its column. This is the correct choice.

If a payoff matrix has a saddle point, it must be both the minimum element of its row and the maximum element of its column. The minmum element of each row is shown in red:

Both saddle point entries are in Column 3, so we select the greater of the two, , as the saddle point entry.

If a payoff matrix has a saddle point, it must be both the minimum element of its row and the maximum element of its column. The minimum element of each row is shown in red:

Of the three, only 100 is the maximum element of its column. This is the saddle point entry.

If a payoff matrix has a saddle point, it must be both the minimum element of its row and the maximum element of its column. The minimum element of each row is shown in red:

Of the three, only 500 is the maximum of its column, making 500 the saddle point entry.

To calculate the mean of the water temperatures first recall the formula for mean.

Now substitute in the values from the problem and solve.

The mean of a data set is the sum of the elements divided by the number of elements. Add the elements, then divide by 7:

.

15 is not the mean.

The median of the data set with an odd number of values is the value that is in the exact center if the elements are arranged in ascending order. The data set is already as such; the center element is 15, as seen below:

That makes 15 the median.

The mode of the data set is the value that appears most frequently. The only value that appears multiple times is 15, as seen below:

.

That makes 15 the mode.

Statements (b) and (c) are correct, but not Statement (a).