What is the probability of guessing a random card?
One of the greatest magic tricks in the world is a magician guessing exactly which card you have thought of! Imagine a cool brown haired guy comes up to you and lays a card down on a table in front of you. He says “think of ANY card” and you say, for example, the two of clubs. He flips the card over on the table that has been right in front of you and it is exactly the card you have named!
In the magic world this has historically been known as a “thought of card” routine, or in some even bolder cases an “any card at any number” routine. The two tricks are statistically exactly the same but nonetheless pretty amazing.
So, what about this trick is so freaking incredible? We can explain that using math, and also show why the same trick shown to children would not be viewed as amazing at all!
Let’s demonstrate this with the two of clubs. There are 52 cards in a shuffled deck of cards. The chances of any particular card being the two of clubs are 1/52, or 1.92%.
Those aren’t very good odds. If a guy ever guesses a card that you merely thought of, give him a standing ovation because it is statistically impossible for that to happen!
But wait! What if he just happened to do this trick for different people over and over and then it happened to work on me! He just has to try 52 times until it works right?
If you were to constantly shuffle a deck of cards, and have someone guess a card you laid down over and over, most people would think that after 52 tries you would be 100% likely to get the card. However, that is not the case! 52 tries puts the probability of guessing correctly at least one time just at 64%!
If you wanted to have an 80% success rate for guessing that random card at least ONE time, you would have to do this trick 80 times to make it very likely that it would work once.
If you were to show this trick to a child under the age of 7, they would not be very impressed. This is because cognitively they have not developed the ability to understand “randomness” and “likeliness” for things like playing cards and mind reading. For example, at one show that I performed in a theater I asked an audience member to think of any city in the world. I had a prediction in the box that guessed cincinatti and I was right!
The audience thought it was awesome, but to the kid sitting next to him, he said “…all you did was write down the city before the show and left it in the box”.
… Yes, isn’t that awesome?!?!
The kid stared at me blankly, and went back to eating popcorn.
The big takeaway is statistically speaking, the world we live in is far more random than we would like to believe, so never take your bets on guessing a card correctly even once at a casino.
Well, unless you are a magician… like me.