Video roulette math puzzle

I recently heard of a game called Netflix roulette where people pick a video at random and watch it regardless of the result.

It got me thinking about probability and I came up with the following problem.

Bob loves the TV show Law & Order. Each day he picks an episode at random and watches it. Given there are 456 episodes of the show, how many days will it take Bob to watch the entire series on average?

Bonus: Figure out a formula for a show that has n episodes.

I will post a complete solution on Thursday. If you figure out the answer, leave a comment on how long it took you to solve it.

Small cases

Consider smaller cases to get an idea.

If a series has just 1 episode, it will take 1 day to watch the entire series.

What about 2 episodes? On the first day, Bob will watch one of the episodes. How long will it take him to watch the remaining episode on average?

We can solve for the number of days N as as a sum of two conditional events. If he picks the episode he has not seen (with probability 0.5), then the conditional expectation is 1 day. If he instead picks the episode he has seen, then he essentially loses a day, and he is back to the starting point–so the expectation is N + 1.

In other words,

N = 1 * Pr(picks episode he has not seen) + (N + 1) * Pr(picks episode he has seen)

N = 1 * 0.5 + (N + 1) * 0.5 = 0.5 N + 1

N = 2

Note that it takes Bob 2 days on average to watch the unique episode that he picks with probability 1/2.

Thus, it takes Bob an average of 3 days (1 day for the first episode, 2 days for the second) to watch a series with two episodes.

Solution

We can think about the problem in terms of rolling a die. Each day Bob picks a new episode randomly is essentially like Bob rolling a die where each face represents an episode number.

The question is: how many times on average must a 6-sided die be rolled until all sides appear at least once?

The first roll can be any of the faces. On the second roll, there are 5 remaining unique faces out of 6. Using the logic above, we can conclude it will take an average of 1 / (5/6) = 6/5 rolls until one sees a different face.

We continue the logic to calculate the number of rolls until a new face. As there are 4 remaining out of 6, this will take 6/4 rolls on average. Continuing the logic, we can conclude the total number of rolls it will take on average to reveal every face at least once is:

1 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 147/10 = 14.7

In other words, for a series with 6 episodes, it will take Bob about 15 days to watch every episode.

For a series with n episodes, the similar series is:

For n = 456, this sum is roughly 3,056.

For very large n, the series sum is roughly n * ln(n)–though this approximation for 456 yields 2792 so it is a very rough approximation.