Three Coin Flips

The following game has a $10 entry fee. You are to flip a fair coin three times. The first time it comes up heads you are paid $5. The second time it comes up heads you're paid an additional $7. The third time it comes up heads you're paid $9 more, for a possible maximum prize of $21. Would you pay the $10 entry fee to play?

If not, what would be a fair price for this game? Click below for the answer.

It's not a good idea to play this game at the offered entry fee. Here are the eight possible outcomes when flipping a coin three times, along with how much you would win after each flip.

When you subtract out the $10 entry fee, you only win "big" ($11) one out of eight times. Three times you win only $2, three times you lose $5, and one out of the eight times you lose your entire $10 entry fee. If you average these, you can expect to lose $1 every time you play. So, if you lower the entry fee to $9 this would be a fair game.

This problem appeared as an exercise in Introductory Graph Theory by Gary Chartrand.
See my Probability GitHub repository for a script that shows how to model this problem in Python.

One equals 0.999...

The following is a mathematical proof that 1 is equal to 0.999.... What's wrong with it? Click below for the answer.

$x = 0.999...$
$10x = 9.999...$
$10x = 9 + 0.999...$
$10x = 9 + x$
$9x = 9$
$x = 1$

There's nothing wrong with it. 1 really is equal to 0.999...

Two equals one?

The following is a mathematical proof that two equals one. What's wrong with it? Click below for the answer.

$a = b$
$aa = ab$
$aa - bb = ab - bb$
$(a + b)(a - b) = b(a - b)$
$a + b = b$
$a + a = a$
$2a = a$
$2 = 1$

The problem is in the fourth step, where both sides of the equation are divided by $(a - b)$. Since $a = b$ is given at the start, $a - b$ is 0, and you can't divide by 0.

State Names

There's only one letter in the English alphabet that is not used in the name of any of the 50 United States. Do you know which letter it is? Click below for the answer.

If you said 'J' or 'Z' you weren't far off. Each of those letters appear only once in the names of the 50 states (thanks to New Jersey and Arizona). The correct answer, though, is the letter Q, which does not appear in any state name.

To find the solution, I used a Python script to load a list of state names and count the occurrence of each letter of the alphabet. This method takes a lot less time than consulting a map.