Probability of an Event

Probability of an Event

When we talk about the probability of an event, we are really asking: "What is the chance that the outcome of an experiment falls into a particular collection of outcomes?" That collection is the event. The definition is straightforward once we have a sample space and a probability assigned to each individual outcome.

Consider the experiment of examining three consecutive pens produced by a machine, classifying each as Good (G, non-defective) or Bad (B, defective). The sample space $S$ has eight equally likely outcomes:

Suppose the probabilities assigned to these outcomes are all equal — each outcome gets probability $\frac{1}{8}$. This is a natural assignment when the machine produces pens independently and each pen is equally likely to be good or bad.

Now define two events:

• Event A: exactly one defective pen.
• Event B: at least two defective pens.

From the sample space:

The probability of an event is simply the sum of the probabilities of all the outcomes that belong to that event. So:

This is the core idea: the probability of an event is the sum of the probabilities of its constituent sample points.

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A Second Example: Unequal Probabilities

Now consider tossing a coin twice. The sample space is:

But this time the probabilities are not equal. Suppose they are assigned as:

Check that these sum to 1:

So this assignment satisfies the axiomatic definition of probability.

Define event $E$: "Both tosses yield the same result." Then:

Define event $F$: "Exactly two heads." Then:

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The General Definition …