Question

we roll a 6-sided die n times. what is the probability that all faces have appeared in some order in some six consecutive rolls? what is the expected number of rolls until such a sequence appears

Answer 1

The probability is 94% and the expected number of rolls is 30.

Step-by-step explanation:

To solve this problem, we will use the concept of coupon collectors’ problem, which deals with the expected number of trials required to collect a complete set of items. In this case, the items are the six faces of the die, and the trials are the rolls of the die.

Let’s assume we roll the die n times. The probability of getting a particular face on any given roll is 1/6. Let P(k) be the probability of getting all six faces in some order in the first k rolls. We can use the coupon collectors’ problem formula to calculate P(k):

P(k) = 1 - (5/6)^k

Now, we want to find the expected number of rolls (E) until all six faces appear in some order in six consecutive rolls. This means we need to find the expected number of rolls until we have a sequence of six rolls with all six faces appearing at least once.

Using the complementary probability method, we can express the probability of not having all six faces in six consecutive rolls after k rolls as:

P(k) = (5/6)^k

The probability of having all six faces in six consecutive rolls after k rolls is:

P(k) = 1 - (5/6)^k

Now, we can find the expected number of rolls (E) using the formula:

E = Σ k * P(k)

where the sum is taken over all k from 1 to infinity.

E = Σ k * (1 - (5/6)^k)

To calculate E, we can use the following identity for the sum of an arithmetic series:

Σ k = n(n + 1)/2

In this case, n = infinity, so the sum becomes:

E = (infinity)(infinity + 1)/2 * (1 - (5/6)^k)

The sum converges to a finite value, which is approximately equal to 11.76. However, since we are dealing with a finite number of faces (6 faces on a die), we need to consider the maximum possible number of rolls, which is n. Therefore, we should calculate the probability of getting all faces in some order in six consecutive rolls within n rolls:

P(n) = 1 - (5/6)ⁿ

For example, if we roll the die 30 times (n=30), the probability of getting all six faces in some order in six consecutive rolls is:

P(30) = 1 - (5/6)³⁰ ≈ 0.94

So, there is approximately a 94% chance of getting all six faces in some order in six consecutive rolls within 30 rolls.

Answer 2

The probability of getting all six faces in some order in six consecutive rolls within 30 rolls is indeed quite high, at around 94%.

Consider the probability of failing to get all faces in six rolls:

The probability of not rolling any one specific face in a single roll is 5/6 (since there are 5 other outcomes).

Therefore, the probability of not getting all six faces in a single set of six rolls is (5/6)^6, which is roughly 0.665.
Calculate the probability of success:

The probability of success is the opposite of the probability of failing for all 30 sets of six rolls.

So, the probability of getting all faces in some order in six consecutive rolls within 30 rolls is 1 - (0.665)^30, which is approximately 0.940 or 94%.