What is Probability?
The probability of an event refers to the
likelihood that the event will occur.
How to Interpret Probability
Mathematically, the probability that an event will occur is expressed
as a number between 0 and 1. Notationally, the probability of event A
is represented by P(A).
- If P(A) equals one, event A will almost definitely occur.
In a
statistical experiment, the sum of probabilities for all possible outcomes
is equal to one. This means, for example, that if an experiment
can have three possible outcomes (A, B, and C), then P(A) + P(B) +
P(C) = 1.
How to Compute Probability: Equally Likely Outcomes
Sometimes, a statistical experiment
can have n possible outcomes, each of which is
equally likely. Suppose a subset of r outcomes are
classified as "successful" outcomes.
The probability that the experiment results in a successful
outcome (S) is:
P(S) = ( Number of successful outcomes ) / ( Total number of equally likely outcomes ) = r / n
Consider the following experiment. An urn has 10 marbles. Two
marbles are red, three are green, and five are blue. If an
experimenter randomly selects 1 marble from the urn, what is the
probability that it will be green?
In this experiment, there are 10 equally likely outcomes, three of
which are green marbles. Therefore, the probability of choosing
a green marble is 3/10 or 0.30.
How to Compute Probability: Law of Large Numbers
One can also think about the probability of an event in terms of its
long-run relative frequency. The relative frequency of
an event is the number of times an event occurs, divided by the
total number of trials.
P(A) = ( Frequency of Event A ) / ( Number of Trials )
For example, suppose 50 customers visit a store each day.
On day 1, 5 out of 50 visitors make a purchase.
Based on data from day 1, the relative frequency of purchase is 5/50 or 0.10.
The next day, 35 out of 50 visitors make a purchase.
Based on data from day 1 and day 2 combined, the relative frequency of purchase is 40/100 or 0.40.
The scatterplot above shows the relative frequency of purchase as
the number of trials (in this case, the number of visitors) increases.
Over many trials, the relative frequency converges toward
a stable value (0.20), which can be interpreted as the probability
that a visitor to the store will make a purchase.
The idea that the relative frequency of an
event will converge on the probability of the event,
as the number of trials increases, is
called the law of large numbers.
Test Your Understanding
Problem
A coin is tossed three times. What is the probability that
it lands on heads exactly one time?
(A) 0.125
(B) 0.250
(C) 0.333
(D) 0.375
(E) 0.500
Solution
The correct answer is (D). If you toss a coin three times, there
are a total of eight possible outcomes. They are: HHH, HHT,
HTH, THH, HTT, THT, TTH, and TTT. Of the eight possible outcomes,
three have exactly one head. They are: HTT, THT, and TTH.
Therefore, the probability that three flips of a coin will
produce exactly one head is 3/8 or 0.375.