What is probability?
Dennis Sun
What is probability?
What does it mean to say that a coin has a 0.5 (or
50%) probability of coming up heads?
It means that if I were to toss a coin many many
times, the fraction of heads will be close to 0.5. The
more tosses, the closer the fraction will be to 0.5.
I performed this experiment.
• When I tossed a coin 10 times, it came up heads 4 times, for a
fraction of 0.4.
• When I tossed a coin 1000 times, it came up heads 514 times, for a
fraction of 0.514.
• When I tossed a coin 10000 times, it came up heads 4955 times, for
a fraction of 0.4955.
Although none of these are exactly 50%, the fraction is clearly
approaching 0.5 as the number of tosses increases.
Frequency Interpretation of Probability
We have just described the interpretation of probability as a
long-run frequency.
Definition
The probability of an event (e.g., “the coin lands heads”) is:
# times the event happens in n replications
n→∞
n
P (event) = lim
• What does it mean to say that the probability is 1?
The event is certain to happen.
• What does it mean to say that the probability is 0?
The event will not happen.
• Probabilities cannot be negative or more than 1.
Problems with the Frequency Interpretation
The frequency interpretation assumes that we can imagine
repeating the phenomenon (e.g., tossing a coin, rolling a die) over
and over.
But in many cases, we only get observe the phenomenon once.
For example, you’ve probably heard weather forecasts like “There’s
a 30% chance of rain tomorrow.”
Frequency Interpretation: “If tomorrow were to happen again
and again, then it would rain in 30% of the tomorrows.”
But tomorrow only happens once. It either rains or it doesn’t. So
what does it mean to say that there’s a 30% chance of rain?
This is a philosophical discussion that we won’t get into in this class.
But it’s important to realize that the frequency interpretation is
not as straightforward as it may first appear.