A bet on a statement P is an arrangement whereby the bettor wins a sum W if P is true and loses a sum L if P is false. Some terminology is helpful:
The expected value of a bet for a bettor is the sum of the quantities
obtained by multiplying the payoff (positive if one wins and negative if
one loses) of each case by the probability of that case.
Example: in the Rams/Saints example, suppose that Pr(P) = 1/3.
Then, he expected values of the bet is $[3 (1/3) - 2 (2/3)], that is, $[1
- 4/3], that is -$1/3.
If one's expected value of a bet is
1.
Consider a bet on P such that I win $2 if P is true and lose $5 is
it's false. Determine, the odds, the stake, the betting quotient,
and what Pr(P) must be for the bet to be fair.
Answer
i. the odds are 5 to 2.
ii. the stake is $7
iii. the betting quotient is 5/7
iv. Pr(P) = 5/7.
2.
You flip a fair coin twice. You win $8 if you get two heads;
$6 if you get only one head, and lose $16 if you get no heads at all.
What's the expected value of the bet?
Answer
Pr(H&H) = 1/2 x 1/2 = 1/4
Pr(only one head) = Pr[(H_{1}&-H_{2}) v (-H_{1}&H_{2})]
= 1/2 x 1/2 + 1/2 x 1/2 = 1/4 + 1/4 = 1/2
Pr(no heads) = Pr (T&T) = 1/4.
So, the expected value is $(8 x 1/4 + 6 x 1/2 - 16 x 1/4) = $(2 + 3
- 4) = $1. So, the bet is adavantangeous.
The Dutch Book
If you make a series of bets such that you lose no matter what, then your bookie has made a Dutch Book against you. It can be shown that a Dutch Book can be made against you just in case the "probabilities" you assign do not obey the rules of probability calculus. Here, instead of a proof, we'll look at an example. Suppose you assign P a "probability" which is less than zero, say -1/2 Then, if W = -$3 and L =$1, the bet will be fair, for -1/2 x -3 = (1+1/2) x 1. But if your wins are negative and your losses positive, you're sure to lose! So, if you're ready to put your money where your mouth is, you'd better know some probability calculus!
Degrees of Belief
Intuitively, one's degree of belief in a statement P is one's level
of certainty that P is true. Obviously, our degree of belief that
2+3 = 5 is very high, but our degree of belief that the Rams will win the
Superbowl is pretty low. The notions developed above help us to measure
exactly one's degree of belief in P is we assume that one is not risk averse,
that is, is ready to bet on the basis of the expected value of a
bet on P. In fact, if we can determine when one judges a bet on P
fair, then we know what one thinks Pr(P) is, since Pr(P) is equal
to the betting quotient, L/(W+L) (WHY?).