Assume that the box contains balls numbered from 1 through 28, and that 3 are selected. A random variable X is defined as 1 times the number of odd balls selected, plus 2 times the number of even.

How many different values are possible for the random variable X? I know that is four. But I cannot fill out the probability density function.

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To fill out the probability density function (PDF) for the random variable X, we need to determine the probability of each possible value. Here's how you can do it:

Step 1: Determine the number of odd and even balls in the box.
- Count the number of odd balls: There are 14 odd balls (numbers: 1, 3, 5, ..., 27).
- Count the number of even balls: There are 14 even balls (numbers: 2, 4, 6, ..., 28).

Step 2: Define the possible values for X.
- X can take values from 0 to 6, as there could be 0 to 3 odd balls selected and 0 to 3 even balls selected.

Step 3: Calculate the probability of each value of X.

For X = 0:
- Probability: P(X = 0) = Probability of selecting 0 odd balls and 3 even balls.
- To calculate this probability, we need to determine the number of ways we can choose 3 even balls out of 14 even balls.
- The formula to calculate this is the combination formula nCr: C(14, 3) = 14! / (3! * (14-3)!).
- Simplifying the expression: C(14, 3) = 14! / (3! * 11!).

For X = 1:
- Probability: P(X = 1) = Probability of selecting 1 odd ball and 2 even balls.
- To calculate this probability, we need to determine the number of ways we can choose 1 odd ball out of 14 odd balls and 2 even balls out of 14 even balls.
- The formula to calculate this is: C(14, 1) * C(14, 2) = 14! / (1! * (14-1)!) * 14! / (2! * (14-2)!).

Similarly, you can calculate the probabilities for X = 2, X = 3, X = 4, X = 5, and X = 6 using the same logic.

The final step is to calculate all the probabilities, which you can do using the combination formula.