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 zero, event A will definitely not occur.
• If P(A) is close to zero, there is only a small chance that event A will occur.
• If P(A) equals 0.5, there is a 50-50 chance that event A will occur.
• If P(A) is close to one, there is a strong chance that event A will occur.
• If P(A) equals one, event A will 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.