the Wealthy Canadian

How to Make 7 Million in 7 Years has an interesting poll going on about what decision you would make if you were on Deal or No Deal, which gave me an idea for a post about expected value.  

Expected value is statistical tool that economists may use to determine what outcome to expect.  It is sum of the probabilities multiplied by the outcomes – note that this is not the same thing as the probable outcome.  This concept can best be described with the use of an example.

Suppose that you and nine friends get together for a game of chance.  Each person bets one dollar and puts their name into a hat.  You draw one name out of the hat and the winner gets to keep all the proceeds.

10 people betting $1 apiece, only one winner. Your odds of winning 1 in 10 or 10% (conversely, your chance of losing is 90%).

Therefore, the probable outcome is that you will not win the draw and thus lose one dollar.

Your expected value is -$1*90% + 10*10% = $0.10 (90% chance of losing your $1 bet, 10% chance of winning $10).

This means that if you kept playing all night you could expect that, on average, you would earn ten cents for every dollar bet.  If you played ten times in a row, statistically speaking, every person should come out even.

When playing at the casino the house generally has the games set up so that the expected outcome results in proceeds to themselves.  For example, according to wikipedia, the game of roulette is set up so that the odds of winning is 1 in 38 and the payout is 35 to 1.

Expected Outcome = (-1$*37/38) + ($35*1/38) = -$0.0526.  Therefore, if I play once then the probability is that I will lose.  If I play all night then I can expect to lose, on average, five cents on every bet.  Thus the house earns five cents on every dollar bet.

Which is a better ‘investment’?  With the draw you expect to earn something every bet whereas with roulette you lose with every bet.

Modern Portfolio Theory uses this same idea to identify the expected value for the return your portfolio will see.