### Beyond AP Statistics

#### Probability Basics

#### Small Samples

#### Distributions

#### Power

### Beyond AP Statistics

#### Probability Basics

#### Small Samples

#### Distributions

#### Power

# 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 almost 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 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, a merchant notices one day that 5 out of 50 visitors to her store make a purchase. The next day, 20 out of 50 visitors make a purchase. The two relative frequencies (5/50 or 0.10 and 20/50 or 0.40) differ. However, summing results over many visitors, she might find that the probability that a visitor makes a purchase gets closer and closer 0.20.

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.

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