A standard deck of 52 cards consists of 4 suites (hearts,...

A standard deck of 52 cards consists of 4 suites (hearts, diamonds, spades and clubs) each containing 13 different values (Ace, $2,3, \ldots,$ $10, \mathrm{~J}, \mathrm{Q}, \mathrm{K})$. If you draw some number of cards at random you might or might not have a pair (two cards with the same value) or three cards all of the same suit. However, if you draw enough cards, you will be guaranteed to have these. For each of the following, find the smallest number of cards you would need to draw to be guaranteed having the specified cards. Prove your answers. (a) Three of a kind (for example, three $7^{\prime}$ s). (b) A flush of five cards (for example, five hearts). (c) Three cards that are either all the same suit or all different suits.

A standard deck of 52 cards consists of 4 suites (hearts, diamonds, spades and clubs) each containing 13 different values (Ace, $2,3, \ldots,$ $10, \mathrm{~J}, \mathrm{Q}, \mathrm{K})$. If you draw some number of cards at random you might or might not have a pair (two cards with the same value) or three cards all of the same suit. However, if you draw enough cards, you will be guaranteed to have these. For each of the following, find the smallest number of cards you would need to draw to be guaranteed having the specified cards. Prove your answers.
(a) Three of a kind (for example, three $7^{\prime}$ s).
(b) A flush of five cards (for example, five hearts).
(c) Three cards that are either all the same suit or all different suits.

Key Concepts

Pigeonhole Principle

This principle states that if n items are placed into m containers, where n > m, then at least one container must contain more than one item. In combinatorial problems, it’s used to prove that a certain configuration must exist once the number of items exceeds a specific threshold, regardless of the selection strategy.

Worst-case Scenario Analysis

This method involves considering the most unfavorable distribution of items to delay a particular outcome for as long as possible. In the context of card problems, worst-case analysis helps determine the minimum number of draws required to force an outcome, ensuring that even in the least favorable circumstances, the desired configuration is achieved.

Combinatorial Reasoning

This concept involves applying counting techniques and logical arguments to ascertain that certain configurations (like matching values or suits) become inevitable past a given point. It relies on systematic evaluation of how items (in this case, cards) can be distributed among various categories under defined constraints.

Transcript

00:02 Okay, i want to get three of a kind from my deck of cards.

00:07 Now i want to know what's the minimum number of cards i can draw to be guaranteed that i get three of a kind.

00:21 So we can figure that out by kind of thinking about the worst case scenario.

00:29 So three of a kind is if i get three of the same number or suit, like the example they gave was like three sevens.

00:42 So let's think about, let's imagine we have really bad luck and we draw an ace, we draw a two, et cetera, all the way up to card number 13, which is a king.

01:04 Okay, so every card we've drawn so far is different and we've drawn 13 cards.

01:11 Well, let's say we do that again.

01:13 We draw another ace, another two, a three, et cetera.

01:24 Well, let's say we've drawn the three and we wouldn't want to draw like say a two because then we'd have three of a kind.

01:35 We wouldn't want to draw an ace or another three because then we'd have three of a kind.

01:40 So we're trying to make our luck as bad as possible.

01:43 We're trying to maximize the number of cards we can draw to look at the worst case scenario.

01:49 So i'm gonna keep drawing different cards that don't give me three of a kind.

01:54 That's gonna take me all the way to king.

01:59 Okay, so at this point i have ace through king.

02:06 I've got two of each card.

02:08 So i've got 26 cards at this point because there's 13 of each number of card.

02:21 Now what happens with my 27th draw? well, it's gotta be a card.

02:27 So it's gotta be an ace, a two, a three, all the way up to king.

02:33 So since i already have two of everything, it's gotta make three of a kind.

02:40 There's no way around it.

02:41 So this 27th card is gonna guarantee that i have three of a kind.

02:50 So 27 is the minimum number of cards i can draw to be guaranteed to have three of a kind.

02:59 There's just no way around it.

03:02 It's gotta happen.

03:05 Okay, b was asking about a flush of five cards.

03:15 For example, five hearts.

03:16 So five of the same suit.

03:24 Five cards, same suit.

03:37 Okay, so let's imagine a scenario for that.

03:40 Let's say i draw, i'll let red represent hearts.

03:48 Let's say i draw four cards that are hearts.

03:54 Well, then i can't draw the fifth card because then my experiment is over...

Please give Ace some feedback