Chapter 4: Probability and Sampling Distributions

Section 4.2: Probability Models

A probability models is a mathematical model for randomness.

Definitions:

• sample space, S or Ω: of a random phenomenon is the set of all possible outcomes.
• event: is any outcome or set of outcomes of a random phenomenon.
• probability model: is a mathematical description of a random phenomenon. The description has two parts:
  • sample space
  • a way of assigning probabilities to events

Probability Rules

1. The probability, P(A) of any event A is 0 ≤ P(A) ≤ 1.
2. P(S) = 1 or P(Ω) = 1.
3. Complement rule: P(A^c) = 1 – P(A). In other words, P(A does not occur) = 1 – P(A).
4. Two events A and B are called disjoint if they have no outcomes in common. If A and B are disjoint,
   then P(A È B) = P(A) + P(B). In other words, P(A or B) = P(A) + P(B).
   In general, if A₁, A₂, ¼, A_k are disjoint, then P(A₁ È A₂ È · · · È A_k) = P(A₁) + P(A₂) + · · · + P(A_k)

The idea of randomness is empirical. In other words, it is based on observation. Probability describes what happens in a large number of trials.

Assigning probabilities: Finite number of outcomes

1. Assign a probability to each individual outcome. These probabilities must be numbers between 0 and 1 and must have sum 1.
2. Probability of an event is the sum of the probabilities of outcomes making up the event.

Assigning probabilities: equally likely outcomes

If an experiment has k possible outcomes, all equally likely, then P(A_k) = 1/k and P(B) = (number of outcomes in B)/k.

Assigning probabilities: Intervals of outcomes

When the sample space has an infinite number of outcomes, then the probability of an event is the area under a density curve. The total area under the density curve is equal to 1.

Normal probability distributions

Recall from section 1.3, the normal distribution is a density curve.

A random variable is a real-valued function defined on the sample space. In other words, it is a variable whose value is a numerical outcome of a random phenomenon. There are two types of random variables:

• X is called a discrete random variable if X has a finite number of possible values
• X is a continuous random variable if X takes all values in an interval of numbers