Random Variables

Question

A number is selected at random from the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. The random variable X is the remainder when the selected number is divided by 3. Find the probability distribution of X. (Enter your probabilities as fractions.)
X    P(X)
0
1
2

Answer 1

The probability is the ratio of successes to the number of possible events.

The problem gives (assume equally likely) the set of events:

            Stated:     1     2     3     4     5     6     7     8     9   10
        N Mod 10:    1     2     0     1     2     0     1     2     0     1 

                              (note: MOD is the remainder function)

So, create a frequency distribution:

    X       Freq(X)          P(X)
    0          3               3/10
    1           4              4/10
    2           3              3/10
                               -------------
                                 10/10       [or P=1]

Answer 2

This is a detailed step-by-step solution:

 

Since there is only a small number of numbers, we can simply count the number of occurences for the remainders. We simply have to take the remainder of each number in the sequence and keep track.

 

The first number is 1, which has a remainder of 1. The next number is 2, with a remainder of 2. The next one is 3, with a remainder of 0, so on and so forth...

 

Doing this for all ten number, we find that four of them have a remainder of 1, three of them have a remainder of 2, and three of them have a remainder of 3.

Since there are ten values, we can get fractions by dividing the number of times each remainder occurs by the total number of times all of the remainders occur. In this case, we get that the probaility of the number having a remainder of 0 is 4/10, the probability of it having a remainder of 1 is 3/10, and the probability of it having a remainder of 2 is 3/10.

 

The reason behind this way of solving the problem is simple: We're just counting the number of times something happens, and taking that over the total number of possibilities. When it comes to probability, this is an extremely common way of solving the problem, especially when it comes to finite mathematics.

 

If you have any questions about this or other finite mathematics inquiries, feel free to ask me.