Since I have not done any mathematical magic posts recently, I wanted to make sure we got one in.
Here is a fun trick you can do for your friends that is all based on simple math. Ask them what the odds would be that if you give them a card in the deck, they could guess the value of the card. They should say one in thirteen, or about eight percent.
What about guessing the value wrong. That would be 92%.
Challenge them to get through the whole entire deck and get every single card’s value wrong. Pretty easy right, you only have an eight percent chance of getting it right. Watch their stunned reaction when they get halfway through and get one correct.
The Method: As I said, it is all based on mathematics. To calculate the odds, you must multiply together the odds of getting it wrong every single time. So, you would have to multiply the .92 by itself fifty-two times. This gives you:
(12/13)^52 = .0183...
Basically, there is a 1.83% chance that you will get through the deck getting it wrong every time. I thought that was pretty cool.
If you’re in a mathematical mood, it gets even cooler. Say you did it with the full card rather than just the value. This would give you:
(51/52)^52 = .3643...
Still good odds. But check this out. Remember back when we studied the number e? Well, what is the reciprocal of e, or 1/e?
1/e = .3678...
They aren’t exact, but that is an extremely close estimate, right! This is true because of our formula for e^x. Let’s go back to it.
e^x ≈ (1 + x/n)^n
In this case, 51/52 would be written as:
(1 + -1/52)^52
With this rule, this should be about e^-1, which means the same thing as the reciprocal of e.
You could use this same logic to figure it out for just the value. (12/13)^52 is the same as:
(1 + -4/52)^52
This means that the odds are approximately e^-4, which is 1/e^4. I thought that this trick is cool, but the reasoning behind it is even cooler.