Saint Petersburg Paradox
Consider the following game:
 a "banker" $B$ plays with a "player" $P$
 $P$ must pay some fixed sum to enter the game
 $B$ flips a coin until TAILS appear  then the game stops
 $B$ pays $2^n$ euro to $P$ if the game stops on $n$th toss
How much a rational should be willing to pay to play the game?
 Expected Value could be a good model to estimate the costs for entering the game
Seq 
Gain 
Prob

$T$ 
$2^1$ 
$0.5$

$HT$ 
$2^1$ 
$0.5^2$

$HHT$ 
$2^2$ 
$0.5^3$

... 
... 
...

$\underbrace{H..H}_{n}T$ 
$2^n$ 
$0.5^n$

Let's calculate the expected value
 $EV = 2 \cdot \cfrac{1}{2} + 2^2 \cdot \cfrac{1}{2^2} + 2^3 \cdot \cfrac{1}{2^3} + ... = \infty$
 so the expected value is $\infty$ with just 50% chance to win only 2 euro
 clearly this is not good enough to model such situation
Sources