If I had to think of one number to represent an entire deck of cards, it would be 6.5.
That’s the approximate average card value in the deck.
Now, as a refresher, to calculate that average, you add up the values of the cards in the deck,
for example:  9 + 5 + 1 + 10 + 6 + 2 + 10 + 7 + 3 + 10 + 8 + 4 + 10, and so on to the end of the deck.
And then, only when you are done summing, do you divide sum by the total number of cards considered, which is 52.
And you should get approximately 6.5.
But if this deck had infinite cards, this method wouldn’t work.
To get any value, we’d have to stop at some point, calculate the average up to that point,
and assume that that partial average is representative of the entire deck.
But would that be a reasonable assumption?
Let’s test it out.
The joker is in here somewhere.
Let’s shuffle the deck and calculate the average of the cards up to the joker.
Now, I want to completely rearrange these cards until the deck is thoroughly randomized.
So let’s use 2 or 3 different shuffling methods.
One way to shuffle is to deal the cards into a few piles.
Another way to shuffle is to do the classic cut shuffle.
Like that.
Okay. Now, the last way we'll shuffle is we'll deal the cards into a lot of piles.
And now the moment of truth.
In this shuffled deck of cards, what will be the average value of the cards up to the joker.
Alright, we've got a 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, joker!
Well, clearly the average of these cards is 10.
Which is very far off from 6.5.
In simply rearranging the cards, did I change the average card value in the deck?
Of course not.
It’s just that a partial average isn’t representative of the entire deck.
The cards that bring down the average are still there, they’re just beyond the joker.
Now, if the joker were further back, we would have a better approximation of the average card value in the deck.
But with an infinite deck, we can’t place the joker at the end, since there is no end.
So how do we calculate the average value of the infinite deck?
This is the same challenge we face when trying to calculate the sum of an infinite series.
So here’s what we do.
We calculate the partial sum and partial average up to a point.
We then calculate the partial sum and partial average up to a further point,
and a further point, and so on, forever, producing infinite partial sums and infinite partial averages.
And if the partial sums and averages converge to a value, we assign that value to the infinite series and infinite deck.
It’s a big task. Do you think it can be completed?
Let me know in the comments and thanks for watching.
