52 card Poisson Distribution experiment?

[mishima]

Hi, I was trying to think of a way to generate a Poisson distribution using a single deck of 52.

Say I am looking at the position of the Ace of spades in the deck after a number of shuffle rounds (1 shuffle round is 7 riffle type shuffles). Success is that an Ace of spades is on top of the deck, failure is that it is not. If the Ace starts in the middle of the deck, finding it on top after 1 shuffle round is extremely unlikely, but with more shuffle rounds that chance increases.

Would that qualify? I'm not sure if its reasonable to assume a mean exists for a given number of shuffles.

If not, what might? I'd just like to devise experiments for all the common distributions using a 52 card deck (binomial of course being clearest).

 

[mathman]

Your example is confusing. Is the Ace of Spades supposed to show up eventually if you keep shuffling or is the question, what happens after n shuffles?

What mean are you asking about in the later question?

 

[mishima]

I'm assuming the Ace of Spades will show up on top with enough shuffling. I want to model the appearance of the Ace of Spades on top using a Poisson distribution and am wondering if that is reasonable.

A common textbook example for Poisson distribution is radioactive decay. A certain substance has a very small chance of decaying in a certain time. I'm trying to make an analogy to this using a deck of cards to experiment with.

The mean in question is one of the requirements for a Poisson distribution (it must be known). For example, in radioactive decay, the average number of decays in a given time period is known. I wasn't sure if the average number of times the Ace comes up on top after a given number of shuffles was knowable.

 

[mathman]

Assuming shuffles are thorough, any particular will show up on top 1/52 times on average. The underlying distribution is binomial, not Poisson.

 

[mfb]

For a proper analysis, you'll need some assumption about the shuffling process. "Completely random arrangement afterwards" is one possible assumption, but probably not a very realistic one.

 

[mishima]

That makes sense. What if the experiment is more like this:

I drop 10 cards from a height of 10 feet. I count the number of times the Ace lays face up in 3 minutes of dropping. Just intuitively, it would seem getting something like 30 times would be more unlikely than 5 times. How is that different from standard examples of a truck passing a certain corner a number of times, or a customer entering a shop a certain number of times?

 

[mfb]

10 cards, including the Ace? Assuming the cards form a proper stack on the ground for some reason: on average it will be on top once every 10 runs. If you drop your cards 50 times in 3 minutes, the expectation value is 5. It could be 3, 8, or something similar, but 30 is very unlikely. You get a binomial distribution with an expectation value of 1/10.