Take a standard deck of 52 playing cards. Shuffle them
thoroughly.
I am going to name 5 cards... let's say the 10, Jack, Queen,
King, and Ace of Spades. 5 out of 52 is just under 10% (9.6%).
I'm going to draw the top 10 cards from the shuffled deck.
10 out of 52 is just under 20% (19.2%).
The question is: How likely is it that at least one of those
10 randomly selected cards is one of the 5 that I preselected?
For comparison, if we reduced the deck to 10 cards, and I
selected only 1 card (that would be 10%), and then I draw 2 cards (which would
be 20% of that deck).
It's a 10% chance for either card; I have 2 cards; thus it
must be 20% chance.
"Ah," you say. "But the second time you draw
you have a 1 in 9 chance of success. 10% for the first try, 11% for the second
try. So, it should be a 21% chance."
Well, that's true, but I only get the higher percent if I
miss on the first pull. It's 1/10 for my first try. 10% of the time I win right
there. Of the remaining 90%, there is a 1 in 9 chance of finding the card. One ninth
of 90% is 10%, so we are actually back at 10% + 10% being 20%.
So we've shown, with only 10% of the deck being a winner,
and pulling only 20% of that deck, I have a 20% chance of winning.
Back to my full deck example with 5 cards and 10 draws. Let's try it twice. Here's the proposal:
You shuffle.
You deal 10 cards.
We see if any of them are the royal flush of spades.
You gather all the cards and shuffle again.
You deal 10 cards.
We see if any of them are the royal flush of spades.
If we miss both times, I give you $10.
If we split, we'll call it even.
If we hit both times, you give me $10.
What do you say? Are you feeling lucky?
