# Calculating the probabilities of different lengths of repetitions of numbers of length 4

I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't total to 1

e.g. Here is my reasoning to calculate the probabilities of the different lengths repetitions for length 4

Probability that there are 0 repeating sequences: e.g. WXYZ 10/10 * (9/10)^3 = 729

Probability that there is 1 repeating sequence of length 2: e.g XXYZ or YXXZ or YZXX 10/10 * (9/10)^2 * 1/10 * 3 = 243

Probability that there is 2 repeating sequence of length 2: e.g XXYY or YYXX 10/10 * 9/10 * (1/10)^2 * 2 = 18

Probability that there is a repeating sequence of length 3: e.g XXXY or YXXX 10/10 * 9/10 * (1/10)^2 * 2 = 18

Probability that there is a repeating sequence of length 4: e.g XXXX 10/10 * (1/10)^2 * 3 = 1

When I add the number of outcomes I get 1009, when I should be getting a 1000. Anyone know what I'm doing wrong?

The third entry in the list has a wrong count. One should not multiply by $2$, since shape XXYY is the same as shape YYXX.
• @BruceTrumbo: Yes, that is also incorrect. And there is mixing of probability and counting, so there are layout problems. And the problem is not fully defined, it is not clear whether initial $0$'s are allowed. Apr 16 '15 at 2:55