Expectation and Expected Value

The word 'expected' is something we talk about in every day conversations. For example, we might say that we expect that the price of a stock will increase. Or we might say that we expect that investment A will make more money than investment B. What we 'expect' is related to, but not exactly the same as the mathematical concept called 'expected value'.

In order to talk about and make decisions using the more
formal, mathematically, rigorous, and precise 'expected value', we need to know
something about probabilities. When we referred to
Morningstar's differentiation of
an investment from a gamble --an *investment* has a positive expected
value while a *gamble* has a negative expected value-- we are
referring to the mathematical concept of expectation. Calculation of the
expected value requires knowledge about probabilities (in the form of
probability distributions), which is at the heart of this study.

The expected value is computed as the sum of the product of the probabilities of all outcomes times the values of those outcomes. Sounds complicated, but it really isn't. Suppose you flip a fair coin and you are paid $2 if it comes up heads, but lose $3 if it comes up tails. Since the probability of a head and a tail are .5 each, the expected value is .5 * 2 + .5 * (-3) or a negative $.5.

Coin flip:

state probability payout product (probability x payout)

------------------------------

heads .5 $2 $1.00

tails .5 -$3
-$1.50

------------------------------

Expected value:

Is this a good 'investment'? Most people would say no and we would concur -- it is, by our definition, a gamble since the expected value is negative.

Suppose you flipped a thumb tack where the probability of its landing point up is .8 and the probability of it landing point down is .2. Suppose you have to pay a fee (premium) of $1 to bet on the flip of the tack, where you will be paid $4 if it lands point down, but nothing if it lands point up. The expected value of the payoff would be .8 * 0 + .2 * .4 = 80cents. Thus, the expected value of the game, subtracting the $1 to play, is a negative 20 cents.

Tac flip:

state probability payout product
(probability x payout)

before

subtracting

fee

------------------------------

point up
.8
$0
$0

point down .2
$4 $0.80

------------------------------

Expected value w/o fee: $0.80

Expected value subtracting fee: -$0.20

This would be a gamble (similar to buying most stock options), not an investment. If you were to play this game over and over again, in the long term, you would most certainly lose. However, if people were offered to play this game only once, some would gladly do so because they have a reasonable chance of winning three times what they wagered, even though the expected value is negative. If the probabilities or payouts were different, the expected value might be positive in which case we would say this is an investment rather than a gamble. Observation: We need to know the probabilities which is the focus of this experiment.