(Simulation: coupon collector's problem) Coupon Collector is...

(Simulation: coupon collector's problem) Coupon Collector is a classic statistics problem with many practical applications. The problem is to pick objects from a set of objects repeatedly and find out how many picks are needed for all the objects to be picked at least once. A variation of the problem is to pick cards from a shuffled deck of 52 cards repeatedly and find out how many picks are needed before you see one of each suit. Assume a picked card is placed back in the deck before picking another. Write a program to simulate the number of picks needed to get four cards, one from each suit and display the four cards picked (it is possible a card may be picked twice). Here is a sample run of the program: Queen of Spades 5 of $\mathrm{Clubs}$ Queen of Hearts 4 of Diamonds Number of picks: 12

(Simulation: coupon collector's problem) Coupon Collector is a classic statistics problem with many practical applications. The problem is to pick objects from a set of objects repeatedly and find out how many picks are needed for all the objects to be picked at least once. A variation of the problem is to pick cards from a shuffled deck of 52 cards repeatedly and find out how many picks are needed before you see one of each suit. Assume a picked card is placed back in the deck before picking another. Write a program to simulate the number of picks needed to get four cards, one from each suit and display the four cards picked (it is possible a card may be picked twice). Here is a sample run of the program:
Queen of Spades
5 of $\mathrm{Clubs}$
Queen of Hearts
4 of Diamonds
Number of picks: 12

Key Concepts

Expected Value

Expected value is a fundamental concept in probability that represents the average outcome of a random process over numerous trials. In problems like the coupon collector's problem, it is used to calculate the average number of selections required to collect all distinct items, providing a benchmark for comparing simulation results.

Coupon Collector's Problem

The coupon collector's problem is a classical probability problem that asks how many random selections are needed to collect all distinct items (coupons) from a set. It demonstrates concepts in probability theory such as expected value and probability distributions, and serves as a model for various real-world collection and sampling problems.

Monte Carlo Simulation

Monte Carlo simulation involves using random sampling to approximate complex probabilistic processes. In the context of the coupon collector's problem, simulations repeatedly perform random selections to estimate the number of picks needed to collect all distinct items, providing insight into the system's behavior over numerous trials.

Random Sampling with Replacement

Random sampling with replacement is a process where each selected item is returned to the original set before the next pick. This technique ensures that the probability distribution remains constant throughout the simulation, making it a key method for modeling problems like the coupon collector's problem where each trial is independent.

Transcript

00:01 Hello, for problem 22, you are going to be choosing cards until you get one card of every suit.

00:14 And then once you do that, you want to display the cards as well as how many picks you needed to make in order to get all four suits.

00:27 So in order to pick cards, i'm going to be.

00:30 Use the random so i'll start by importing random and then instead of setting up a deck of cards i separated out rank and suit so rank here is just the a list including all of the ranks of cards in a typical deck and then suit is a list with all of the suits for those cards i'm going to start two empty lists one for cards and one for card suits just to make it easy to check whether or not we've gotten all four card suits.

01:08 I'm also going to start two counters.

01:11 One is n, which is the number of suits, and one which is p, which is the number of picks that we've made.

01:20 So both of those will start with zero.

01:23 And we're going to use a while loop.

01:26 So while the number of suits is less than four, because we want all four suits.

01:32 While the number of suits is less than four, we're going to choose a card at random.

01:37 So card rank will be random .choice rank and card suit will be random.

01:44 Choice of suit.

01:45 If card suit is in our list of card suits, then we know that it's not really helpful to us because it's a repeat suit.

01:58 So we'll increase our pick number because we've picked another card, but you could just not leave anything, but with while loops, i think that it's really important to make sure that the counter that the while is working on is being updated regularly...

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