Problem abstraction

A standard deck of $52$ cards has $26$ red cards: it has $13$ hearts, $13$ diamonds, as well as $26$ black cards ($13$ spades, as well as $13$ clubs). Let us draw $5$ cards from the deck at once, and return those cards to the deck afterward.

What is the expected number of draws before we see all $26$ red cards?

#Use case Let us say that there is a set of $N = 100$ cards in a game. $M = 30$ cards are of rare rarity and $N - M = 70$ cards are of common rarity. We buy booster packs of size $= 10$. The question is: how many booster packs need to be bought to collect all $M = 30$ cards?

#Attempted solution I have managed to calculate the approximate number of booster packs necessary to get $M = 30$ rare cards by calculating the expectation of the above hypergeometric distribution ($\mu$) and then calculating $M/\mu$. However, this is not the correct solution since it does not take into account the possibility of collecting duplicates.

Regarding the Coupon collector's problem, I'm not sure if it is applicable since we always draw a single coupon, whereas in my use case a booster pack contains more than a single card.

Related: https://stats.stackexchange.com/questions/198915/in-the-coupon-collectors-problem-with-group-drawings-why-does-the-probability

Simulations

Problem Abstraction

$10^6$ trials were conducted, AVG: $38.947$, STDEV: $12.3653$ draws

Use case

$10^6$ trials were conducted, AVG: $38.535$, STDEV: $11.962$ draws

Problem abstraction

A standard deck of $52$ cards has $26$ red cards: it has $13$ hearts, $13$ diamonds, as well as $26$ black cards ($13$ spades, as well as $13$ clubs). Let us draw $5$ cards from the deck at once, and return those cards to the deck afterward.

What is the expected number of draws before we see all $26$ red cards?

Use case

Let us say that there is a set of $N = 100$ cards in a game. $M = 30$ cards are of rare rarity and $N - M = 70$ cards are of common rarity. We buy booster packs of size $= 10$. The question is: how many booster packs need to be bought to collect all $M = 30$ cards?

Attempted solution

I have managed to calculate the approximate number of booster packs necessary to get $M = 30$ rare cards by calculating the expectation of the above hypergeometric distribution ($\mu$) and then calculating $M/\mu$. However, this is not the correct solution since it does not take into account the possibility of collecting duplicates.

Regarding the Coupon collector's problem, I'm not sure if it is applicable since we always draw a single coupon, whereas in my use case a booster pack contains more than a single card.

Related: https://stats.stackexchange.com/questions/198915/in-the-coupon-collectors-problem-with-group-drawings-why-does-the-probability

Simulations

Problem Abstraction

$10^6$ trials were conducted, AVG: $38.947$, STDEV: $12.3653$ draws

Use case

$10^6$ trials were conducted, AVG: $38.535$, STDEV: $11.962$ draws

Problem abstraction

A standard deck of $52$ cards has $26$ red cards: it has $13$ hearts, $13$ diamonds, as well as $26$ black cards ($13$ spades, as well as $13$ clubs). Let us draw $5$ cards from the deck at once, and return those cards to the deck afterward.

What is the expected number of draws before we see all $26$ red cards?

#Use case Let us say that there is a set of $N = 100$ cards in a game. $M = 30$ cards are of rare rarity and $N - M = 70$ cards are of common rarity. We buy booster packs of size $= 10$. The question is: how many booster packs need to be bought to collect all $M = 30$ cards?

#Attempted solution I have managed to calculate the approximate number of booster packs necessary to get $M = 30$ rare cards by calculating the expectation of the above hypergeometric distribution ($\mu$) and then calculating $M/\mu$. However, this is not the correct solution since it does not take into account the possibility of collecting duplicates.

Regarding the Coupon collector's problem, I'm not sure if it is applicable since we always draw a single coupon, whereas in my use case a booster pack contains more than a single card.

Related: https://stats.stackexchange.com/questions/198915/in-the-coupon-collectors-problem-with-group-drawings-why-does-the-probability

Simulations

Problem Abstraction

$10^6$ trials were conducted, AVG: $194.57$, STDEV: $61.83$ draws

Use case

$10^6$ trials were conducted, AVG: $385.90$, STDEV: $119.72$ draws

Problem abstraction

A standard deck of $52$ cards has $26$ red cards: it has $13$ hearts, $13$ diamonds, as well as $26$ black cards ($13$ spades, as well as $13$ clubs). Let us draw $5$ cards from the deck at once, and return those cards to the deck afterward.

What is the expected number of draws before we see all $26$ red cards?

#Use case Let us say that there is a set of $N = 100$ cards in a game. $M = 30$ cards are of rare rarity and $N - M = 70$ cards are of common rarity. We buy booster packs of size $= 10$. The question is: how many booster packs need to be bought to collect all $M = 30$ cards?

#Attempted solution I have managed to calculate the approximate number of booster packs necessary to get $M = 30$ rare cards by calculating the expectation of the above hypergeometric distribution ($\mu$) and then calculating $M/\mu$. However, this is not the correct solution since it does not take into account the possibility of collecting duplicates.

Regarding the Coupon collector's problem, I'm not sure if it is applicable since we always draw a single coupon, whereas in my use case a booster pack contains more than a single card.

Related: https://stats.stackexchange.com/questions/198915/in-the-coupon-collectors-problem-with-group-drawings-why-does-the-probability

Simulations

Problem Abstraction

$10^6$ trials were conducted, AVG: $38.947$, STDEV: $12.3653$ draws

Use case

$10^6$ trials were conducted, AVG: $38.535$, STDEV: $11.962$ draws

Problem abstraction

A standard deck of $52$ cards has $26$ red cards: it has $13$ hearts, $13$ diamonds, as well as $26$ black cards ($13$ spades, as well as $13$ clubs). Let us draw $5$ cards from the deck at once, and return those cards to the deck afterward.

What is the expected number of draws before we see all $26$ red cards?

#Use case Let us say that there is a set of $N = 100$ cards in a game. $M = 30$ cards are of rare rarity and $N - M = 70$ cards are of common rarity. We buy booster packs of size $= 10$. The question is: how many booster packs need to be bought to collect all $M = 30$ cards?

#Attempted solution I have managed to calculate the approximate number of booster packs necessary to get $M = 30$ rare cards by calculating the expectation of the above hypergeometric distribution ($\mu$) and then calculating $M/\mu$. However, this is not the correct solution since it does not take into account the possibility of collecting duplicates.

Regarding the Coupon collector's problem, I'm not sure if it is applicable since we always draw a single coupon, whereas in my use case a booster pack contains more than a single card.

Related: https://stats.stackexchange.com/questions/198915/in-the-coupon-collectors-problem-with-group-drawings-why-does-the-probability

Simulations

Problem Abstraction

$10^6$ trials were conducted, AVG: $194.57$, STDEV: $61.83$ draws

Use case

Problem Abstraction

$10^6$ trials were conducted, AVG: $385.90$, STDEV: $119.72$ draws

Problem abstraction

A standard deck of $52$ cards has $26$ red cards: it has $13$ hearts, $13$ diamonds, as well as $26$ black cards ($13$ spades, as well as $13$ clubs). Let us draw $5$ cards from the deck at once, and return those cards to the deck afterward.

What is the expected number of draws before we see all $26$ red cards?

#Use case Let us say that there is a set of $N = 100$ cards in a game. $M = 30$ cards are of rare rarity and $N - M = 70$ cards are of common rarity. We buy booster packs of size $= 10$. The question is: how many booster packs need to be bought to collect all $M = 30$ cards?

#Attempted solution I have managed to calculate the approximate number of booster packs necessary to get $M = 30$ rare cards by calculating the expectation of the above hypergeometric distribution ($\mu$) and then calculating $M/\mu$. However, this is not the correct solution since it does not take into account the possibility of collecting duplicates.

Regarding the Coupon collector's problem, I'm not sure if it is applicable since we always draw a single coupon, whereas in my use case a booster pack contains more than a single card.

Related: https://stats.stackexchange.com/questions/198915/in-the-coupon-collectors-problem-with-group-drawings-why-does-the-probability

Simulations

Problem Abstraction

$10^6$ trials were conducted, AVG: $194.57$, STDEV: $61.83$ draws

Use case

$10^6$ trials were conducted, AVG: $385.90$, STDEV: $119.72$ draws