Topic 2.4 Notes – Introduction to Probability

What Probability Is

A random process is something you can repeat where the result of one trial is uncertain, even though the possible results are known ahead of time. A coin toss, a die roll, or classifying a trip as delayed or not delayed all fit.

• An outcome is one result of one trial.
• The sample space $S$ is the full set of all possible outcomes.
• An event $E$ is any collection of outcomes from that sample space.

For probability models, two facts always have to hold:

$$0 \le P(E) \le 1$$

$$P(S)=1$$

That means:

• probability 0 = impossible
• probability 1 = certain
• every trial ends in one and only one sample-space outcome
• a valid model has outcome probabilities that are nonnegative and add to 1

This sounds simple, but a lot of mistakes later come from building the sample space badly.

Building the Sample Space and Event

Your sample space has to be exhaustive and nonoverlapping.

• Exhaustive means every possible result is listed.
• Nonoverlapping means one trial can’t land in two listed outcomes at once.

If you toss a coin twice and care about order, the sample space is:

$$S=\{HH, HT, TH, TT\}$$

A tree diagram is a quick way to see all four ordered outcomes.

Why are $HT$ and $TH$ separate? Because they are different ordered outcomes.

If the event is “exactly one head,” then

$$E=\{HT,TH\}$$

A few things to keep straight:

• a single outcome could be $\{HH\}$
• a whole event could contain several outcomes, like $\{HT,TH\}$
• an impossible event contains no outcomes at all

Common traps:

• Using labels like “even” and “greater than 3” as sample-space outcomes for a die roll. Those overlap.
• Naming categories and acting like they must be equally likely. They might not be.

Finding Probabilities from a Model

The first question is whether the outcomes are equally likely.

If they are, use theoretical probability:

$$P(E)=\frac{\text{number of outcomes in }E}{\text{number of outcomes in }S}$$

For “exactly one head” in two fair tosses:

• favorable outcomes = $HT, TH$ so 2
• total outcomes = $HH, HT, TH, TT$ so 4

$$P(E)=\frac{2}{4}=\frac{1}{2}$$

If outcomes are not equally likely, you add the probabilities of the outcomes in the event.

Example. Suppose trip status has probabilities:

• no delay = 0.72
• moderate delay = 0.21
• severe delay = ?

Since probabilities in the sample space add to 1,

$$P(\text{severe})=1-0.72-0.21=0.07$$

Then “some delay” means moderate or severe:

$$0.21+0.07=0.28$$

The big mistake here is dividing by the number of categories just because you listed categories. Getting a number between 0 and 1 does not prove your method was right.

Complements and Not E

The complement of $E$ means “not $E$.” You may see $E'$, $\overline{E}$, or $E^C$.

$$P(E^C)=1-P(E)$$

$$P(E)+P(E^C)=1$$

In the diagram, focus on the top row. One panel shows the event $E$, and the other shows everything outside that event, which is the complement.

Event and complement

This is useful when “not $E$” is easier to count.

Example. In two fair tosses, let $E$ = “at least one head.”
Its complement is “no heads,” which is just $TT$.

$$P(E)=1-P(TT)=1-\frac14=\frac34$$

Translate wording carefully:

• “at least one” ↔ none
• “at least $k$” ↔ fewer than $k$
• “more than $k$” ↔ at most $k$
• “at most $k$” ↔ more than $k$

Boundary trap students miss all the time: the complement of “at most 4” is “greater than 4,” not “at least 4.”

Interpreting Probability in Context

Probability means long-run relative frequency. If a model says $P(\text{some delay})=0.28$, that means over many trips under the same conditions, about 28% would have some delay.

It does not mean exactly 28 of the next 100 trips will be delayed. Short runs can bounce around a lot.

Also separate these two ideas:

• theoretical probability comes from a model
• empirical relative frequency comes from actual data or simulation

On the exam, write the event clearly, show the setup, give the probability, and if asked, interpret it in context.