Non-Decreasing

Suppose you roll a fair die 10 times. What is the probability that the sequence of rolls is non-decreasing (i.e., each subsequent roll is never less than the previous one)?

Answer

To find the probability of obtaining a non-decreasing sequence in 10 rolls, we count the number of ways to distribute 10 indistinguishable rolls into 6 distinguishable bins (representing the number of 1's, 2's, ..., 6's). This is done using the stars and bars method.

The number of ways to place 5 bars among 10 stars is:

\binom{10 + 5}{5} = \binom{15}{5} = 3003

For each of these combinations, there is only one way to arrange it in a non-decreasing way. On the other hand, the total number of possible sequences is $6^{10}$ .

Thus, the probability is:

P = \frac{\binom{15}{5}}{6^{10}} = \frac{3003}{60466176} \approx 0.000049664

Notes and comments

Comment 1: The generalized formula for the probability of $$ n $$ non-decreasing throws in a 6-sided fair die is:

P_n = \frac{\binom{n + 5}{5}}{6^n}

This formula represents the number of ways to distribute $$ n $$ indistinguishable objects (rolls) into 6 distinguishable bins (die faces).

Comment 2: If we would have non-decreasing OR non-increasing then we would multiply by two.