You have an interesting deck of cards:
there are an unknown number of different types of cards,
each card type is repeated many times (let's say infinitely),
if these cards are placed face down, how many must you flip over before you are fairly confident you have found all the different types of card in the deck?
"Unknown number of different types" lead to unknown answer.
I'm not sure how to interpret the answer above...
let's say there are 6 different types of cards and an infinite number of each type,
how many cards do you need to look at to be 95% confident there are not 7 different types?
To determine the minimum number of cards you need to flip over before being fairly confident you have found all the different types of cards, we can use a statistical approach called the Coupon Collector's Problem.
The Coupon Collector's Problem is a classic problem in probability theory that can be used to estimate the expected number of trials needed to collect a set of distinct items. In this case, each card type represents a distinct item.
Let's denote "n" as the number of different types of cards in the deck. According to the Coupon Collector's Problem, the expected number of cards you need to flip over is given by:
E(n) = n * (1 + 1/2 + 1/3 + ... + 1/n)
This equation represents the harmonic sum of 1 + 1/2 + 1/3 + ... + 1/n, which approximates the average number of trials that need to be done in order to collect all the different card types.
For example, let's say there are 10 different types of cards in the deck. Plugging this value into the equation, we get:
E(10) = 10 * (1 + 1/2 + 1/3 + ... + 1/10) ≈ 29.29
This means that, on average, you would need to flip over approximately 29 cards before being confident that you have found all 10 different card types.
Note that this is an expected value, and there is still some degree of uncertainty involved. The actual number of cards required can vary, but as you continue to flip over more cards, the probability of finding all the different card types increases.
Therefore, in order to be fairly confident you have found all the different types of cards, you may need to flip over more cards than the expected value, depending on the specific randomness of the deck.