What Are the Odds?
Elementary Probability
Sandra LaFave
West Valley College
What
does it mean to say there’s a certain probability that an event will
occur, or that “the odds” favor or oppose a certain event?
It
depends whether you mean
·
Classical
Probability,
or
·
Relative
Frequency,
or
·
Subjective
Probability
I. Classical Probability
Also
called the a priori theory of probability, it is associated with the
mathematicians Pascal and Fermat.
Paradigm:
card games in which the deck contains a fixed number of cards
The probability (P) of an
event (a) equals the number of possible favorable outcomes (f) divided
by the total number of possible outcomes (n). Or,
P(a)
= f / n
The classical theory assumes
that all possible outcomes are equally likely, and that we know n. For
example, we know the number of cards in a deck of regular cards (n =
52, for the first draw), and we assume that the deck is fair, i.e., that every
card has an equal chance of being drawn. What is the probability that I’ll draw
an ace, if there are four aces in the deck?
P(a)
= 4/52, or 1/13, or .0769 or 7.69%
Once
I’ve drawn a card, and it’s, say, not an ace, the probability that the next
draw yields an ace goes up, since n has gone down.
P(a)
= 4/51, or .0784 or 7.84%
As
n gets smaller and no aces are drawn, the probability of an ace on the
next draw increases dramatically.
If
all the possible outcomes are known and equally likely, then classical probability
yields a definite number every time. It doesn’t generalize; its answer is quite
specific. Classical probability doesn’t concern itself with sample size or
representativeness.
The
results of classical probability obey the Law of Large Numbers. That
is, the results will be closer and closer to the percentages predicted the more
often we carry out the draw. If we draw the first card from a fair deck one
million times, we’ll find that we will get an ace in very close to 7.7% of the
cases. This is why casinos consistently make a profit on the games they offer;
they know the odds.
II.
The Gambler’s Fallacy
Suppose you’re playing a
game in which all outcomes are equally likely (e.g., rolling dice), and you
are on a losing streak. You commit the Gambler’s Fallacy if you believe your
losing streak makes it more likely that you’ll roll the numbers you want
on the next roll (because you’re “due”). The truth is that your odds don’t change;
you start over with each roll.
III. Relative Frequency
We
need to use relative frequency probability when the number of possible outcomes
(n) is so large we can’t observe all of them. For example, what is the
probability that a 60yearold woman will die of a heart attack within ten years?
We can’t observe all 60yearold women.
But
we can predict the probability for random typical women who resemble one
another,
if we observe enough of them. Here is where we need to watch for the fallacy
of hasty generalization, because size and representativeness of the sample really
matter here. We need to make sure we observe enough instances (n
should
be large enough), and the ones we observe need to be typical.
P(a)
= observed f / observed n
The probability of 60yearold
women dying of a heart attack within ten years (a) equals the number of 60
yearolds
who do die of a heart attack within ten years (f), divided by all comparable
60yearold women (n). So if we observed 1000 60yearold women over
ten years, and observed that 35 of them died of heart attacks, we could say
the probability of a random 60yearold woman dying of a heart attack is around
3.5% (35/1000).
P(a)
= 35 / 1000
There’s
always the chance that our numbers are unreliable because our sample is
unrepresentative.
But these results are very reliable if the sample is representative and over
1500 (2% margin of error).
Naturally,
we can’t predict with certainty for any particular woman. If we could, we wouldn’t
need probability theory.
IV. Subjective Probability
This
has more to do with sociology and salesmanship than with statistics. What odds
will you take? Those are the odds you get on the subjectivist theory. It’s called
subjectivist because it relies only on what people are willing to bet. What
odds are offered? Whatever the parties agree to; this isn’t an objective matter.
The odds you get when you bet at the racetrack are of this type. The people
who run the track make odds they think the customers will find attractive enough
to bet on. People who follow racing closely think they are knowledgeable about
the horses, but any horse in the race can win. (If the outcome were based on
objective probabilities, then a smart player could win consistently.) But the
track and the OTB want to make money, don’t they? If you’re willing to bet on
a long shot, you can take your chances.
So
what kind of probability applies to winning the lottery? To whether the Giants
will win the next World Series?
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